INTERNATIONAL GONGRESS -stability involves a strengthened notion called reduced uniform K-stability, which matches the reduced coercivity in (1.14) (see [19,29,39][19,29,39][19,29,39][19,29,39][19,29,39] ). Recall that T~T~tilde(T)\tilde{\mathbb{T}}T~ denotes a maximal torus of Aut(X,L)Autâ¡(X,L)Aut(X,L)\operatorname{Aut}(X, L)Autâ¡(X,L), and N~QN~Qtilde(N)_(Q)\tilde{N}_{\mathbb{Q}}N~Q is defined similar to (1.3).
Definition 2.3. A polarized manifold (X,L)(X,L)(X,L)(X, L)(X,L) is uniformly K-stable (resp. reduced uniformly K-stable) if there exists γ>0γ>0gamma > 0\gamma>0γ>0 such that any test configuration (X,L)(X,L)(X,L)(\mathcal{X}, \mathscr{L})(X,L) satisfies MNA(X,L)≥MNA(X,L)≥M^(NA)(X,L) >=\mathbf{M}^{\mathrm{NA}}(\mathcal{X}, \mathscr{L}) \geqMNA(X,L)≥γ⋅JNA(X,L)(γ⋅JNA(X,L)gamma*J^(NA)(X,L)(:}\gamma \cdot \mathbf{J}^{\mathrm{NA}}(\mathcal{X}, \mathscr{L})\left(\right.γ⋅JNA(X,L)( resp. MNA(X,L)≥γ⋅infξ∈N~QJNA(Xξ,Lξ))MNA(X,L)≥γ⋅infξ∈N~Q JNAXξ,Lξ{:M^(NA)(X,L) >= gamma*i n f_(xi in tilde(N)_(Q))J^(NA)(X_(xi),L_(xi)))\left.\mathbf{M}^{\mathrm{NA}}(\mathcal{X}, \mathscr{L}) \geq \gamma \cdot \inf _{\xi \in \tilde{N}_{\mathbb{Q}}} \mathbf{J}^{\mathrm{NA}}\left(\mathcal{X}_{\xi}, \mathscr{L}_{\xi}\right)\right)MNA(X,L)≥γ⋅infξ∈N~QJNA(Xξ,Lξ))
Here the twist (Xξ,Lξ)Xξ,Lξ(X_(xi),L_(xi))\left(\mathcal{X}_{\xi}, \mathscr{L}_{\xi}\right)(Xξ,Lξ) is introduced by Hisamoto [39]. One way to define it as a test configuration is by resolving the composition of birational morphisms (X,L)→(XC=(X,L)→XC=(X,L)rarr(X_(C)=:}(\mathcal{X}, \mathscr{L}) \rightarrow\left(X_{\mathbb{C}}=\right.(X,L)→(XC=X×C,LC=p1∗L)→σξ(XC,LC)X×C,LC=p1∗L→σξXC,LC{:X xxC,L_(C)=p_(1)^(**)L)rarr"sigma_(xi)"(X_(C),L_(C))\left.X \times \mathbb{C}, L_{\mathbb{C}}=p_{1}^{*} L\right) \xrightarrow{\sigma_{\xi}}\left(X_{\mathbb{C}}, L_{\mathbb{C}}\right)X×C,LC=p1∗L)→σξ(XC,LC) where σξσξsigma_(xi)\sigma_{\xi}σξ is the C∗C∗C^(**)\mathbb{C}^{*}C∗-action generated by ξξxi\xiξ. Alternatively, it can be defined in a more general setting of filtrations (see Example 2.8).
2.2. Non-Archimedean pluripotential theory
We discuss how non-Archimedean pluripotential theory as developed by Boucksom-Jonsson can be applied to study K-stability. Corresponding to a regularization result in the complex analytic case, an u.s.c. function ϕ:XNA→R∪{+∞}Ï•:XNA→R∪{+∞}phi:X^(NA)rarrRuu{+oo}\phi: X^{\mathrm{NA}} \rightarrow \mathbb{R} \cup\{+\infty\}Ï•:XNA→R∪{+∞} is called a nonArchimedean psh potential if it is a decreasing limit of a sequence from HNAHNAH^(NA)\mathscr{H}^{\mathrm{NA}}HNA. Denote the space of such functions by PSHNAPSHNAPSH^(NA)\mathrm{PSH}^{\mathrm{NA}}PSHNA. Boucksom-Jonsson introduced the following nonArchimedean version of the finite energy space. First corresponding to (1.10), for any ϕ∈ϕ∈phi in\phi \inϕ∈PSHNAPSHNAPSH^(NA)\mathrm{PSH}^{\mathrm{NA}}PSHNA, define
This space is again equipped with a strong topology which makes ENAENAE^(NA)\mathbf{E}^{\mathrm{NA}}ENA continuous. Boucksom-Jonsson showed in [22] that the non-Archimedean Monge-Ampère measure MANA(ϕ)MANA(Ï•)MA^(NA)(phi)\mathrm{MA}^{\mathrm{NA}}(\phi)MANA(Ï•) is well defined for any ϕ∈(E1)NAϕ∈E1NAphi in(E^(1))^(NA)\phi \in\left(\mathcal{E}^{1}\right)^{\mathrm{NA}}ϕ∈(E1)NA such that if {ϕk}k∈N⊂HNAÏ•kk∈N⊂HNA{phi_(k)}_(k inN)subH^(NA)\left\{\phi_{k}\right\}_{k \in \mathbb{N}} \subset \mathscr{H}^{\mathrm{NA}}{Ï•k}k∈N⊂HNA converges to ϕÏ•phi\phiÏ• strongly, then MANA(ϕk)MANAÏ•kMA^(NA)(phi_(k))\mathrm{MA}^{\mathrm{NA}}\left(\phi_{k}\right)MANA(Ï•k) converges to MANA(ϕ)MANA(Ï•)MA^(NA)(phi)\mathrm{MA}^{\mathrm{NA}}(\phi)MANA(Ï•) weakly.
A large class of potentials come from filtrations (see [19]). Set Rm=H0(X,mL)Rm=H0(X,mL)R_(m)=H^(0)(X,mL)R_{m}=H^{0}(X, m L)Rm=H0(X,mL).
Definition 2.4. A filtration is the data F={FλRm⊆Rm;λ∈R,m∈N}F=FλRm⊆Rm;λ∈R,m∈NF={F^(lambda)R_(m)subeR_(m);lambda inR,m inN}\mathcal{F}=\left\{\mathcal{F}^{\lambda} R_{m} \subseteq R_{m} ; \lambda \in \mathbb{R}, m \in \mathbb{N}\right\}F={FλRm⊆Rm;λ∈R,m∈N} that satisfies the following four conditions:
(i) FλRm⊆Fλ′RmFλRm⊆Fλ′RmF^(lambda)R_(m)subeF^(lambda^('))R_(m)\mathcal{F}^{\lambda} R_{m} \subseteq \mathcal{F}^{\lambda^{\prime}} R_{m}FλRm⊆Fλ′Rm, if λ≥λ′λ≥λ′lambda >= lambda^(')\lambda \geq \lambda^{\prime}λ≥λ′; (ii) FλRm=⋂λ′<λFλ′RmFλRm=⋂λ′<λ Fλ′RmF^(lambda)R_(m)=nnn_(lambda^(') < lambda)F^(lambda^('))R_(m)\mathcal{F}^{\lambda} R_{m}=\bigcap_{\lambda^{\prime}<\lambda} \mathcal{F}^{\lambda^{\prime}} R_{m}FλRm=⋂λ′<λFλ′Rm;
(iii) FλRm⋅Fλ′Rm′⊆Fλ+λ′Rm+m′FλRmâ‹…Fλ′Rm′⊆Fλ+λ′Rm+m′F^(lambda)R_(m)*F^(lambda^('))R_(m^('))subeF^(lambda+lambda^('))R_(m+m^('))\mathscr{F}^{\lambda} R_{m} \cdot \mathcal{F}^{\lambda^{\prime}} R_{m^{\prime}} \subseteq \mathcal{F}^{\lambda+\lambda^{\prime}} R_{m+m^{\prime}}FλRmâ‹…Fλ′Rm′⊆Fλ+λ′Rm+m′, for λ,λ′∈Rλ,λ′∈Rlambda,lambda^(')inR\lambda, \lambda^{\prime} \in \mathbb{R}λ,λ′∈R and m,m′∈Nm,m′∈Nm,m^(')inNm, m^{\prime} \in \mathbb{N}m,m′∈N;
(iv) There exist e−,e+∈Ze−,e+∈Ze_(-),e_(+)inZe_{-}, e_{+} \in \mathbb{Z}e−,e+∈Z such that Fme−Rm=RmFme−Rm=RmF^(me)_(-)R_(m)=R_(m)\mathcal{F}^{m e}{ }_{-} R_{m}=R_{m}Fme−Rm=Rm and Fme+Rm=0Fme+Rm=0F^(me_(+))R_(m)=0\mathcal{F}^{m e_{+}} R_{m}=0Fme+Rm=0 for m∈m∈m inm \inm∈Z≥0Z≥0Z_( >= 0)\mathbb{Z}_{\geq 0}Z≥0.
Filtration FFF\mathscr{F}F is finitely generated if its extended Rees algebra R(F)R(F)R(F)\mathscr{R}(\mathscr{F})R(F) is finitely generated where
In this case FFF\mathcal{F}F induces a degeneration of XXXXX into X0=Proj(⨁m,λFλRm/F>λRm)X0=Projâ¡â¨m,λ FλRm/F>λRmX_(0)=Proj(bigoplus_(m,lambda)F^(lambda)R_(m)//F > lambdaR_(m))\mathcal{X}_{0}=\operatorname{Proj}\left(\bigoplus_{m, \lambda} \mathcal{F}^{\lambda} R_{m} / \mathcal{F}>\lambda R_{m}\right)X0=Projâ¡(â¨m,λFλRm/F>λRm).
For a general F,{FλRℓ;λ∈R}F,FλRâ„“;λ∈RF,{FlambdaR_(â„“);lambda inR}\mathscr{F},\left\{\mathcal{F} \lambda R_{\ell} ; \lambda \in \mathbb{R}\right\}F,{FλRâ„“;λ∈R} generates a filtration F(ℓ)F(â„“)F(â„“)\mathscr{\mathscr { F }}(\ell)F(â„“) on R(ℓ):=⨁m∈NRmℓR(â„“):=â¨m∈N Rmâ„“R^((â„“)):=bigoplus_(m inN)R_(mâ„“)R^{(\ell)}:=\bigoplus_{m \in \mathbb{N}} R_{m \ell}R(â„“):=â¨m∈NRmâ„“, which induces a non-Archimedean psh potential ϕˇ(ℓ)∈HNAϕˇ(â„“)∈HNAphi^(ˇ)^((â„“))inH^(NA)\check{\phi}^{(\ell)} \in \mathscr{H}^{\mathrm{NA}}ϕˇ(â„“)∈HNA. Define
ϕF=(lim supℓ→+∞ϕˇ(ℓ))∗Ï•F=lim supℓ→+∞ ϕˇ(â„“)∗phi_(F)=(l i m   s u p_(â„“rarr+oo)phi^(ˇ)^((â„“)))^(**)\phi_{\mathcal{F}}=\left(\limsup _{\ell \rightarrow+\infty} \check{\phi}^{(\ell)}\right)^{*}Ï•F=(lim supℓ→+∞ϕˇ(â„“))∗
where (⋅)∗(â‹…)∗(*)^(**)(\cdot)^{*}(â‹…)∗ denotes the upper-semicontinuous regularization.
Example 2.5. Filtration FFF\mathscr{F}F is a ZZZ\mathbb{Z}Z-filtration if FλRm=F⌈λ⌉RmFλRm=F⌈λ⌉RmF^(lambda)R_(m)=F|~lambda~|R_(m)\mathscr{F}^{\lambda} R_{m}=\mathscr{F}\lceil\lambda\rceil R_{m}FλRm=F⌈λ⌉Rm. By [19,63,69][19,63,69][19,63,69][19,63,69][19,63,69], there is a one-to-one correspondence between test configurations equipped with relatively ample QQQ\mathbb{Q}Q polarizations and finitely generated ZZZ\mathbb{Z}Z-filtrations. Any test configuration (X,L)(X,L)(X,L)(\mathcal{X}, \mathscr{L})(X,L) defines such a filtration by
Conversely, if FFF\mathscr{F}F is a finitely generated ZZZ\mathbb{Z}Z-filtration, then (X:=ProjC[t](R(Fˇ(ℓ))),1ℓOX(1))X:=ProjC[t]â¡(R(Fˇ(â„“))),1â„“OX(1)(X:=Proj_(C[t])(R((F^(ˇ))(â„“))),(1)/(â„“)O_(X)(1))\left(\mathcal{X}:=\operatorname{Proj}_{\mathbb{C}[t]}(\mathcal{R}(\check{\mathcal{F}}(\ell))), \frac{1}{\ell} \mathcal{O}_{X}(1)\right)(X:=ProjC[t]â¡(R(Fˇ(â„“))),1â„“OX(1)) is a test configuration for ℓâ„“â„“\ellâ„“ sufficiently divisible.
Example 2.6. In Definition 2.1 of test configurations, if we do not require LLL\mathscr{L}L to be πÏ€pi\piÏ€ semiample, then we call (X,L)(X,L)(X,L)(\mathcal{X}, \mathscr{L})(X,L) a model (of (X×C,p1∗L)X×C,p1∗L(X xxC,p_(1)^(**)L)\left(X \times \mathbb{C}, p_{1}^{*} L\right)(X×C,p1∗L) ). The same definition in (2.7) defines a filtration also denoted by F(x,L)F(x,L)F_((x,L))\mathscr{F}_{(x, \mathscr{L})}F(x,L). However, in general the filtration is not finitely generated anymore. Fix any model (X,L)(X,L)(X,L)(\mathcal{X}, \mathscr{L})(X,L) such that L¯L¯bar(L)\overline{\mathscr{L}}L¯ is big over X¯X¯bar(X)\bar{X}X¯ (we call such (X,L)(X,L)(X,L)(\mathcal{X}, \mathscr{L})(X,L) a big model for (X,L)(X,L)(X,L)(X, L)(X,L) ). In [46] we obtained the following formula for the non-Archimedean Monge-Ampère measure of ϕ=ϕ(X,L):=ϕF(X,L)Ï•=Ï•(X,L):=Ï•F(X,L)phi=phi_((X,L)):=phi_(F_((X,L)))\phi=\phi_{(X, \mathscr{L})}:=\phi_{\mathcal{F}_{(X, \mathscr{L})}}Ï•=Ï•(X,L):=Ï•F(X,L) which generalizes (2.5):
Example 2.7. Any v∈XQdiv v∈XQdiv v inX_(Q)^("div ")v \in X_{\mathbb{Q}}^{\text {div }}v∈XQdiv defines a filtration: for any λ∈Rλ∈Rlambda inR\lambda \in \mathbb{R}λ∈R and m∈Z≥0m∈Z≥0m inZ_( >= 0)m \in \mathbb{Z}_{\geq 0}m∈Z≥0, define
Boucksom-Jonsson proved in [21] that MANA(ϕFv)=V⋅δvMANAÏ•Fv=V⋅δvMA^(NA)(phi_(F_(v)))=V*delta_(v)\mathrm{MA}^{\mathrm{NA}}\left(\phi_{\mathcal{F}_{v}}\right)=\mathbf{V} \cdot \delta_{v}MANA(Ï•Fv)=V⋅δv.
Generalizing the case of test configurations, Boucksom-Jonsson showed that the non-Archimedean functionals from (2.2)-(2.4) are well defined for all ϕ∈(E1)NAϕ∈E1NAphi in(E^(1))^(NA)\phi \in\left(\mathcal{E}^{1}\right)^{\mathrm{NA}}ϕ∈(E1)NA by using integrals over XNAXNAX^(NA)X^{\mathrm{NA}}XNA mentioned before (for example, for HNAHNAH^(NA)\mathbf{H}^{\mathrm{NA}}HNA use (2.6)).
Example 2.9. For any filtration FFF\mathcal{F}F, it is known that ϕF∈(E1)NAÏ•F∈E1NAphi_(F)in(E^(1))^(NA)\phi_{\mathcal{F}} \in\left(\mathcal{E}^{1}\right)^{\mathrm{NA}}Ï•F∈(E1)NA. Following [19], define
Similar to Theorem 1.2, we also have important regularization properties:
Theorem 2.10 ([22]). For any ϕ∈(E1)NAϕ∈E1NAphi in(E^(1))^(NA)\phi \in\left(\mathcal{E}^{1}\right)^{\mathrm{NA}}ϕ∈(E1)NA, there exists {ϕk}k∈N⊂HNAÏ•kk∈N⊂HNA{phi_(k)}_(k inN)subH^(NA)\left\{\phi_{k}\right\}_{k \in \mathbb{N}} \subset \mathscr{H}^{\mathrm{NA}}{Ï•k}k∈N⊂HNA (i.e., ϕk=ϕ(Xk,Lk)Ï•k=Ï•Xk,Lkphi_(k)=phi_((X_(k),L_(k)))\phi_{k}=\phi_{\left(X_{k}, \mathscr{L}_{k}\right)}Ï•k=Ï•(Xk,Lk) for a test configuration (Xk,Lk))Xk,Lk{:(X_(k),L_(k)))\left.\left(\mathcal{X}_{k}, \mathscr{L}_{k}\right)\right)(Xk,Lk)) such that ϕk→ϕÏ•k→ϕphi_(k)rarr phi\phi_{k} \rightarrow \phiÏ•k→ϕ in the strong topology and FNA(ϕk)→FNAÏ•k→F^(NA)(phi_(k))rarr\mathbf{F}^{\mathrm{NA}}\left(\phi_{k}\right) \rightarrowFNA(Ï•k)→FNA(ϕ)FNA(Ï•)F^(NA)(phi)\mathbf{F}^{\mathrm{NA}}(\phi)FNA(Ï•) for F∈{E,Λ,EKX}F∈E,Λ,EKXFin{E,Lambda,E^(K_(X))}\mathbf{F} \in\left\{\mathbf{E}, \boldsymbol{\Lambda}, \mathbf{E}^{K_{X}}\right\}F∈{E,Λ,EKX}.
Boucksom-Jonsson conjectured that the same conclusion should also hold for HNAHNAH^(NA)\mathbf{H}^{\mathrm{NA}}HNA. This conjecture is still open in general and it is important in the non-Archimedean approach to the YTD conjecture. We have made progress in this direction.
Theorem 2.11 ([45,46]). (1) For any ϕ∈(E1)NAϕ∈E1NAphi in(E^(1))^(NA)\phi \in\left(\mathcal{E}^{1}\right)^{\mathrm{NA}}ϕ∈(E1)NA, there exist models {(Xk,Lk)}k∈NXk,Lkk∈N{(X_(k),L_(k))}_(k inN)\left\{\left(\mathcal{X}_{k}, \mathscr{L}_{k}\right)\right\}_{k \in \mathbb{N}}{(Xk,Lk)}k∈N such that ϕk=ϕ(xk,Lk)→ϕÏ•k=Ï•xk,Lk→ϕphi_(k)=phi_((x_(k),L_(k)))rarr phi\phi_{k}=\phi_{\left(x_{k}, \mathscr{L}_{k}\right)} \rightarrow \phiÏ•k=Ï•(xk,Lk)→ϕ in the strong topology and HNA(ϕk)→HNA(ϕ)HNAÏ•k→HNA(Ï•)H^(NA)(phi_(k))rarrH^(NA)(phi)\mathbf{H}^{\mathrm{NA}}\left(\phi_{k}\right) \rightarrow \mathbf{H}^{\mathrm{NA}}(\phi)HNA(Ï•k)→HNA(Ï•).
(2) For any big model (X,L)(X,L)(X,L)(\mathcal{X}, \mathscr{L})(X,L), we have the following formula that generalizes (2.4):
The idea for proving the first statement is similar to the Archimedean setting in [8]. First we regularize the measure MANA(ϕ)MANA(Ï•)MA^(NA)(phi)\mathrm{MA}^{\mathrm{NA}}(\phi)MANA(Ï•) with converging entropy. In fact, we find a way to regularize it by using measures supported at finitely many points in XQdiv XQdiv X_(Q)^("div ")X_{\mathbb{Q}}^{\text {div }}XQdiv . Then we use the solution of non-Archimedean Monge-Ampère equations obtained in [18] to get the wanted potentials which are known to be associated to models. However, in the non-Archimedean case, there is not yet a characterization of measures associated to test configurations which prevents us from regularizing via test configurations. The second statement in Theorem 2.11 follows from the formula (2.8), and it prompts us to propose the following algebro-geometric conjecture which would strengthen the classical Fujita approximation theorem.
Conjecture 2.12. Let X¯X¯bar(X)\overline{\mathcal{X}}X¯ be a smooth (n+1)(n+1)(n+1)(n+1)(n+1)-dimensional projective variety. Let L¯L¯bar(L)\bar{L}L¯ be a big line bundle over X¯X¯bar(X)\bar{X}X¯. Then there exist birational morphisms μk:X¯k→X¯Î¼k:X¯k→X¯mu_(k): bar(X)_(k)rarr bar(X)\mu_{k}: \bar{X}_{k} \rightarrow \bar{X}μk:X¯k→X¯ and decompositions μk∗L¯=L¯k+Ekμk∗L¯=L¯k+Ekmu_(k)^(**) bar(L)= bar(L)_(k)+E_(k)\mu_{k}^{*} \overline{\mathscr{L}}=\overline{\mathscr{L}}_{k}+E_{k}μk∗L¯=L¯k+Ek in N1(X¯)QN1(X¯)QN^(1)( bar(X))_(Q)N^{1}(\overline{\mathcal{X}})_{\mathbb{Q}}N1(X¯)Q with L¯kL¯kbar(L)_(k)\overline{\mathcal{L}}_{k}L¯k semiample and EkEkE_(k)E_{k}Ek effective such that
It is easy to show that this conjecture is true if L¯L¯bar(L)\overline{\mathscr{L}}L¯ admits a birational Zariski decomposition. The author verified this conjecture for certain examples of big line bundles due to Nakamaya which do not admit such decompositions (see [46]). Y. Odaka observed that when (X,L)(X,L)(X,L)(\mathcal{X}, \mathscr{L})(X,L) is a big model for a polarized spherical manifold (for example, a polarized toric manifold), X¯X¯bar(X)\bar{X}X¯ is a Mori dream space which implies that L¯L¯bar(L)\overline{\mathscr{L}}L¯ admits a Zariski decomposition and hence the above conjecture holds true.
2.3. Stability of Fano varieties
In this section, we assume that XXXXX is a QQQ\mathbb{Q}Q-Fano variety (i.e., −KX−KX-K_(X)-K_{X}−KX is an ample QQQ\mathbb{Q}Q line bundle and XXXXX has at worst klt singularities). Corresponding to (1.9), we have a nonArchimedean D functional. For general test configurations, it first appeared in Berman's work [4] and was reformulated in [19] using non-Archimedean potentials:
The notions of Ding-stability and uniform Ding-stability are defined if MNAMNAM^(NA)\mathbf{M}^{\mathrm{NA}}MNA is replaced by DNADNAD^(NA)\mathbf{D}^{\mathrm{NA}}DNA in Definitions 2.2 and 2.3. In general, we have the inequality MNA(X,L)≥DNA(X,L)MNA(X,L)≥DNA(X,L)M^(NA)(X,L) >= D^(NA)(X,L)\mathbf{M}^{\mathrm{NA}}(\mathcal{X}, \mathscr{L}) \geq \mathbf{D}^{\mathrm{NA}}(\mathcal{X}, \mathscr{L})MNA(X,L)≥DNA(X,L). For Fano varieties, special test configurations play important roles. A test configuration (X,L)(X,L)(X,L)(\mathcal{X}, \mathscr{L})(X,L) is called special if the central fiber X0X0X_(0)\mathcal{X}_{0}X0 is a QQQ\mathbb{Q}Q-Fano variety and L=−KX/P1L=−KX/P1L=-K_(X//P^(1))\mathscr{L}=-K_{X / \mathbb{P}^{1}}L=−KX/P1. For special test configurations, we have DNA=MNA=−ENA=DNA=MNA=−ENA=D^(NA)=M^(NA)=-E^(NA)=\mathbf{D}^{\mathrm{NA}}=\mathbf{M}^{\mathrm{NA}}=-\mathbf{E}^{\mathrm{NA}}=DNA=MNA=−ENA= : Fut x0(ξ)x0(ξ)x_(0)(xi)x_{0}(\xi)x0(ξ), the last quantity being the Futaki invariant on X0X0X_(0)\mathcal{X}_{0}X0 for the holomorphic vector field ξξxi\xiξ that generates the C∗C∗C^(**)\mathbb{C}^{*}C∗-action. The importance of special test configurations was first pointed out in Tian's work [64] motivated by compactness results from metric geometry. The following results show their importance from the point of view of algebraic geometry:
Theorem 2.13 ([35,44,52], see also [7,21])[7,21])[7,21])[7,21])[7,21]). For any QQQ\mathbb{Q}Q-Fano variety, KKKKK-stability is equivalent to Ding-stability, and they are equivalent to KKKKK-stability or Ding-stability over special test configurations. Moreover, the same conclusion holds true if stability is replaced by semistability, polystability, or reduced uniform stability.
The proofs of these results depend on a careful process of Minimal Model Program first used in [52] to transform any given test configuration into a special one. Moreover, crucial calculations show that the relevant invariants such as MNAMNAM^(NA)\mathbf{M}^{\mathrm{NA}}MNA or DNADNAD^(NA)\mathbf{D}^{\mathrm{NA}}DNA decrease along the MMP process. Theorem 2.13 leads directly to a valuative criterion for K-stability. To state it, first define for any v∈XQdiv v∈XQdiv v inX_(Q)^("div ")v \in X_{\mathbb{Q}}^{\text {div }}v∈XQdiv an invariant (see Example 2.9):
Let T~T~tilde(T)\tilde{\mathbb{T}}T~ be a maximal torus of Aut(X)Autâ¡(X)Aut(X)\operatorname{Aut}(X)Autâ¡(X) and (XQdiv )T~XQdiv T~(X_(Q)^("div "))^( tilde(T))\left(X_{\mathbb{Q}}^{\text {div }}\right)^{\tilde{\mathbb{T}}}(XQdiv )T~ be the set of T~T~tilde(T)\tilde{\mathbb{T}}T~-invariant divisorial valuations. Define the following invariant ((ξ,v)↦vξ(ξ,v)↦vξ((xi,v)|->v_(xi):}\left((\xi, v) \mapsto v_{\xi}\right.((ξ,v)↦vξ is the action appeared in Example 2.8):
δ(X)=infv∈XQdivAX(v)SX(v),δT~(X)=infv∈(XQdiv)supξ∈N~RAX(vξ)SX(vξ)δ(X)=infv∈XQdiv AX(v)SX(v),δT~(X)=infv∈XQdiv supξ∈N~R AXvξSXvξdelta(X)=i n f_(v inX_(Q)^(div))(A_(X)(v))/(S_(X)(v)),quaddelta_( tilde(T))(X)=i n f_(v in(X_(Q)^(div)))s u p_(xi in tilde(N)_(R))(A_(X)(v_(xi)))/(S_(X)(v_(xi)))\delta(X)=\inf _{v \in X_{\mathbb{Q}}^{\mathrm{div}}} \frac{A_{X}(v)}{S_{X}(v)}, \quad \delta_{\tilde{\mathbb{T}}}(X)=\inf _{v \in\left(X_{\mathbb{Q}}^{\mathrm{div}}\right)} \sup _{\xi \in \tilde{\mathbb{N}}_{\mathbb{R}}} \frac{A_{X}\left(v_{\xi}\right)}{S_{X}\left(v_{\xi}\right)}δ(X)=infv∈XQdivAX(v)SX(v),δT~(X)=infv∈(XQdiv)supξ∈N~RAX(vξ)SX(vξ)
Here we use the convention that AX(vtriv )/SX(vtriv )=+∞AXvtriv /SXvtriv =+∞A_(X)(v_("triv "))//S_(X)(v_("triv "))=+ooA_{X}\left(v_{\text {triv }}\right) / S_{X}\left(v_{\text {triv }}\right)=+\inftyAX(vtriv )/SX(vtriv )=+∞ for the trivial valuation vtriv vtriv v_("triv ")v_{\text {triv }}vtriv .
Theorem 2.14. The following statements are true.
(1) ([35,42])X([35,42])X([35,42])X([35,42]) X([35,42])X is KKKKK-semistable if δ(X)≥1δ(X)≥1delta(X) >= 1\delta(X) \geq 1δ(X)≥1.
(2) ([34,35])X([34,35])X([34,35])X([34,35]) X([34,35])X is uniformly KKKKK-stable if and only if δ(X)>1δ(X)>1delta(X) > 1\delta(X)>1δ(X)>1.
(3) ( [15,35,42])X[15,35,42])X[15,35,42])X[15,35,42]) X[15,35,42])X is KKKKK-stable if and only if AX(v)>S(v)AX(v)>S(v)A_(X)(v) > S(v)A_{X}(v)>S(v)AX(v)>S(v) for any nontrivial v∈XQdiv v∈XQdiv v inX_(Q)^("div ")v \in X_{\mathbb{Q}}^{\text {div }}v∈XQdiv .
(4) ([44]) XXXXX is reduced uniformly KKKKK-stable if and only if δT~(X)>1δT~(X)>1delta_( widetilde(T))(X) > 1\delta_{\widetilde{\mathbb{T}}}(X)>1δT~(X)>1.
To get these, we first use the fact as pointed out in [19] that for a special test configuration (X,L)(X,L)(X,L)(\mathcal{X}, \mathscr{L})(X,L), the valuation ord X0X0X_(0)\mathcal{X}_{0}X0 of the function field C(X×C)C(X×C)C(X xxC)\mathbb{C}(X \times \mathbb{C})C(X×C) restricts to become a divisorial valuation v∈XQdiv v∈XQdiv v inX_(Q)^("div ")v \in X_{\mathbb{Q}}^{\text {div }}v∈XQdiv . A crucial observation is then made in [42]: F(x,L)λRm=F(x,L)λRm=F_((x,L))^(lambda)R_(m)=\mathcal{F}_{(x, \mathscr{L})}^{\lambda} R_{m}=F(x,L)λRm=Fvλ+mAX(v)Rm(Fvλ+mAX(v)RmF_(v)^(lambda+mA_(X)(v))R_(m)(:}\mathscr{F}_{v}^{\lambda+m A_{X}(v)} R_{m}\left(\right.Fvλ+mAX(v)Rm( see (2.9)). This implies vol(F(X,L)(t))=vol(Fv(t+AX(v)))volâ¡F(X,L)(t)=volâ¡Fvt+AX(v)vol(F_((X,L))^((t)))=vol(F_(v)^((t+A_(X)(v))))\operatorname{vol}\left(\mathscr{F}_{(X, \mathscr{L})}^{(t)}\right)=\operatorname{vol}\left(\mathcal{F}_{v}^{\left(t+A_{X}(v)\right)}\right)volâ¡(F(X,L)(t))=volâ¡(Fv(t+AX(v))), which, together with (2.10), leads to 22^(2){ }^{2}2
This, together with Theorem 2.13, already gives the sufficient condition for the K-(semi)stability. The criterion for uniform K-stability follows from a similar argument and K. Fujita's inequality, 1nS(v)≤JNA(Fv)≤nS(v)1nS(v)≤JNAFv≤nS(v)(1)/(n)S(v) <= J^(NA)(F_(v)) <= nS(v)\frac{1}{n} S(v) \leq \mathbf{J}^{\mathrm{NA}}\left(\mathcal{F}_{v}\right) \leq n S(v)1nS(v)≤JNA(Fv)≤nS(v) [34]. For reduced uniform stability, another identity AX(vξ)−S(vξ)=AX(v)−S(v)+AXvξ−Svξ=AX(v)−S(v)+A_(X)(v_(xi))-S(v_(xi))=A_(X)(v)-S(v)+A_{X}\left(v_{\xi}\right)-S\left(v_{\xi}\right)=A_{X}(v)-S(v)+AX(vξ)−S(vξ)=AX(v)−S(v)+ Fut X(ξ)X(ξ)_(X)(xi)_{X}(\xi)X(ξ) proved in [44] is needed.
As we will see in Section 3.2, a main reason for introducing the (reduced) uniform K-stability is that it is much easier to use in making connection with the (reduced) coercivity in the complex analytic setting. On the other hand, we now have the following fundamental result:
Theorem 2.15 ([56]). Let XXXXX be a QQQ\mathbb{Q}Q-Fano variety. Then XXXXX is KKKKK-stable if and only if XXXXX is uniformly KKKKK-stable. More generally, XXXXX is reduced uniformly stable if and only if XXXXX is KKKKK polystable. Moreover, these statements hold true for any log Fano pair.
This is achieved by several works. First, according to a work of Blum-Liu-Xu [13], divisorial valuations on XXXXX associated to special test configurations are log canonical places of complements. By deep boundedness of Birkar and Haccon-McKernan-Xu, it was also shown that there exists a quasimonomial valuation (i.e., a monomial valuation on a smooth birational model) that achieves the infimum defining δ(X)δ(X)delta(X)\delta(X)δ(X) (and a similar result holds more generally for δT~(X))δT~(X){:delta_( tilde(T))(X))\left.\delta_{\tilde{\mathbb{T}}}(X)\right)δT~(X)). Then the main problem becomes proving a finite generation property for the minimizing valuation, which is achieved by using deep techniques from birational
2 The original argument in [42] also explicitly relates the filtration F(X,L)F(X,L{:F_(()X,L)\left.\mathcal{F}_{(} \mathcal{X}, \mathscr{L}\right)F(X,L) to a filtration of the section ring of X0X0X_(0)\mathcal{X}_{0}X0 induced by the C∗C∗C^(**)\mathbb{C}^{*}C∗-action.
algebraic geometry in [56]. In fact, in the past several years, the algebraic study of K-stability for Fano varieties has flourished, and there are many important results which answer fundamental questions in this subject. We highlight two such achievements:
(1) Algebraic construction of projective moduli space of K-polystable Fano varieties. This is achieved in a collection of works, settling different issues in the construction including boundedness, separatedness, properness, and projectivity. Moreover, concrete examples of compact moduli spaces have been identified. We refer to [56,70][56,70][56,70][56,70][56,70] for extensive discussions on related topics.
(2) Fujita-Odaka [36] introduced quantizations of the δ(X)δ(X)delta(X)\delta(X)δ(X) invariant: for each m∈Nm∈Nm inNm \in \mathbb{N}m∈N,
δm(X)=inf{lct(X,D):D is of m-basis type }δm(X)=inf{lctâ¡(X,D):D is of m-basis type }delta_(m)(X)=i n f{lct(X,D):D" is of "m"-basis type "}\delta_{m}(X)=\inf \{\operatorname{lct}(X, D): D \text { is of } m \text {-basis type }\}δm(X)=inf{lctâ¡(X,D):D is of m-basis type }
where DDDDD is of mmmmm-basis type if D=1mNm∑i=1Nm{si=0}D=1mNm∑i=1Nm si=0D=(1)/(mN_(m))sum_(i=1)^(N_(m)){s_(i)=0}D=\frac{1}{m N_{m}} \sum_{i=1}^{N_{m}}\left\{s_{i}=0\right\}D=1mNm∑i=1Nm{si=0} where {si}si{s_(i)}\left\{s_{i}\right\}{si} is a basis of H0(X,mL)H0(X,mL)H^(0)(X,mL)H^{0}(X, m L)H0(X,mL). Blum-Jonsson [12] proved limm→+∞δm(X)=δ(X)limm→+∞ δm(X)=δ(X)lim_(m rarr+oo)delta_(m)(X)=delta(X)\lim _{m \rightarrow+\infty} \delta_{m}(X)=\delta(X)limm→+∞δm(X)=δ(X). This provides a practical tool to verify uniform stability of Fano varieties. Ahmadinezhad-Zhuang [1] further introduced new techniques for estimating the δmδmdelta_(m)\delta_{m}δm and δδdelta\deltaδ invariant which lead to many new examples of K-stable Fano varieties. All of these culminate in the recent determination of deformation types of smooth Fano threefolds that contain K-polystable ones (see [3]).
The techniques developed in the study of (weighted) K-stability of Fano varieties have also been applied to treat an optimal degeneration problem that is motivated by the Hamilton-Tian conjecture in differential geometry (see [74] for background of this conjecture). This is formulated as a minimization problem for valuations in [38] which defines the following functional (cf. (2.12) and (2.11)), for any valuation v∈XNAv∈XNAv inX^(NA)v \in X^{\mathrm{NA}}v∈XNA,
Combined with previous works, the uniqueness part in particular confirms a conjecture of Chen-Sun-Wang about the algebraic uniqueness of limits under normalized KählerRicci flows on Fano manifolds (see [62]).
2.4. Normalized volume and local stability theory of klt singularities
A similar minimization problem for valuations was actually studied even earlier in the local setting, which motivates the formulation and the proof of Theorem 2.16. Let (X,x)(X,x)(X,x)(X, x)(X,x) be a klt singularity. Denote by ValX,xValX,xVal_(X,x)\mathrm{Val}_{X, x}ValX,x the space of real valuations that have center xxxxx. The following normalized volume functional was introduced in [43]: for any v∈ValX,xv∈ValX,xv inVal_(X,x)v \in \operatorname{Val}_{X, x}v∈ValX,x,
Here AX(v)AX(v)A_(X)(v)A_{X}(v)AX(v) is again the logloglog\loglog discrepancy functional and vol(v)volâ¡(v)vol(v)\operatorname{vol}(v)volâ¡(v) is defined as
The expression in (2.13) is inspired by the work of Martelli-Sparks-Yau [57] on a volume minimization property of Reeb vector fields associated to Ricci-flat Kähler cone metrics. In [43] we started to consider the minimization of vol over ^ValX,x vol over ^ValX,xwidehat(" vol over ")Val_(X,x)\widehat{\text { vol over }} \operatorname{Val}_{X, x} vol over ^ValX,x and define the invariant vol^(X,x)=infv∈ValX,xvol^(v)vol^(X,x)=infv∈ValX,x vol^(v)widehat(vol)(X,x)=i n f_(v inVal_(X,x)) widehat(vol)(v)\widehat{\operatorname{vol}}(X, x)=\inf _{v \in \mathrm{Val}_{X, x}} \widehat{\operatorname{vol}}(v)vol^(X,x)=infv∈ValX,xvol^(v). We proved that the invariant vol^(X,x)vol^(X,x)widehat(vol)(X,x)\widehat{\operatorname{vol}}(X, x)vol^(X,x) is strictly positive and further conjectured the existence, uniqueness of minimizing valuations which should have finite generated associated graded rings. For a concrete example, it was shown by the author and Y. Liu that for an isolated quotient singularity X=Cn/Γ,vol^(Cn/Γ,0)=nn|Γ|X=Cn/Γ,vol^Cn/Γ,0=nn|Γ|X=C^(n)//Gamma, widehat(vol)(C^(n)//Gamma,0)=(n^(n))/(|Gamma|)X=\mathbb{C}^{n} / \Gamma, \widehat{\operatorname{vol}}\left(\mathbb{C}^{n} / \Gamma, 0\right)=\frac{n^{n}}{|\Gamma|}X=Cn/Γ,vol^(Cn/Γ,0)=nn|Γ| and the exceptional divisor of the standard blowup obtains the infimum.
This minimization problem was proposed by the author to attack a conjecture of Donaldson-Sun, which states that the metric tangent cone at any point on a GromovHausdorff limit of Kähler-Einstein manifolds depends only on the algebraic structure (see [62]). This conjecture has been confirmed in a series of following-up papers [51, 53, 54]. Algebraically, we have the following results regarding this minimization problem.
Theorem 2.17. (1) There exists a valuation that achieves the infimum in defining vol^(X,x)vol^(X,x)widehat(vol)(X,x)\widehat{\operatorname{vol}}(X, x)vol^(X,x). Moreover, this minimizing valuation is quasimonomial and unique up to rescaling.
(2) A divisorial valuation v∗v∗v_(**)v_{*}v∗ is the minimizer if and only if it is the exceptional divisor of a plt blowup and also the associated log Fano pair is KKKKK-semistable.
The first statement is a combination of works by Harold Blum, Chenyang Xu, and Ziquan Zhuang [11,71,72] partly based on calculations from [42,43]. The second statement was proved in Li−XuLi−XuLi-Xu\mathrm{Li}-\mathrm{Xu}Li−Xu [54] (see also [11]) by extending the global argument from [52] to the local case, and it shows a close relationship between the local and global theory. In fact, it is in proving the affine cone case of this statement when valuative criterion for KKK\mathrm{K}K-(semi)stability
was first discovered in [42]. A similar statement is true for more general quasimonomial minimizing valuations [53]. However, the finite generation conjecture from [43] is still open in general, and seems to require deeper boundedness property of Fano varieties.
We also like to mention that Yuchen Liu obtained a surprising local-to-global comparison inequality by generalizing an estimate of Kento Fujita:
Theorem 2.18 ([55]). For any closed point xxxxx on a KKKKK-semistable QQQ\mathbb{Q}Q-Fano variety XXXXX, we have
For example, if x∈Xx∈Xx in Xx \in Xx∈X is a regular point, (2.14) recovers Fujita's beautiful inequality, namely (−KX)⋅n≤(n+1)n−KXâ‹…n≤(n+1)n(-K_(X))^(*n) <= (n+1)^(n)\left(-K_{X}\right)^{\cdot n} \leq(n+1)^{n}(−KX)â‹…n≤(n+1)n for any K-semistable XXXXX [33]. Inequality (2.14) has applications in controlling singularities on the varieties that correspond to boundary points of moduli spaces. In order for this to be effective, good estimates of vol^(X,x)vol^(X,x)widehat(vol)(X,x)\widehat{\operatorname{vol}}(X, x)vol^(X,x) for klt singularities need to be developed. In particular, it is still interesting to understand better the vol invariants and ^ vol invariants and ^widehat(" vol invariants and ")\widehat{\text { vol invariants and }} vol invariants and ^ associated minimizers for 3-dimensional klt singularities. For more discussion on related topics, we refer to the survey [48].
3. ARCHIMEDEAN (COMPLEX ANALYTIC) THEORY VS. NON-ARCHIMEDEAN THEORY
3.1. Correspondence between Archimedean and non-Archimedean objects
In this section, we discuss results showing a general philosophy that non-Archimedean objects encode the information of corresponding Archimedean objects "at infinity."
Let (X,L)(X,L)(X,L)(\mathcal{X}, \mathscr{L})(X,L) be a test configuration and h~h~tilde(h)\tilde{h}h~ be a smooth psh metric on LLL\mathscr{L}L. Via the isomorphism (X,L)×CC∗≅X×C∗(X,L)×CC∗≅X×C∗(X,L)xx_(C)C^(**)~=X xxC^(**)(\mathcal{X}, \mathscr{L}) \times_{\mathbb{C}} \mathbb{C}^{*} \cong X \times \mathbb{C}^{*}(X,L)×CC∗≅X×C∗, we get a path Φ~={φ~(s)}s∈RΦ~={φ~(s)}s∈Rwidetilde(Phi)={ tilde(varphi)(s)}_(s inR)\widetilde{\Phi}=\{\tilde{\varphi}(s)\}_{s \in \mathbb{R}}Φ~={φ~(s)}s∈R of smooth ω0ω0omega_(0)\omega_{0}ω0-psh potentials where s=−log|t|2s=−logâ¡|t|2s=-log |t|^(2)s=-\log |t|^{2}s=−logâ¡|t|2. With these notation, we have the following slope formula:
Theorem 3.1 ([20,59,64,67]). The slope at infinity of a functional F∈{E,Λ,I,J,M}F∈{E,Λ,I,J,M}Fin{E,Lambda,I,J,M}\mathbf{F} \in\{\mathbf{E}, \boldsymbol{\Lambda}, \mathbf{I}, \mathbf{J}, \mathbf{M}\}F∈{E,Λ,I,J,M} is given by the corresponding non-Archimedean functional
There is a more canonical analytic object associated to a test configuration. Recall from section 1.5 that by a geodesic ray Φ={φ(s)}s∈[0,+∞)Φ={φ(s)}s∈[0,+∞)Phi={varphi(s)}_(s in[0,+oo))\Phi=\{\varphi(s)\}_{s \in[0,+\infty)}Φ={φ(s)}s∈[0,+∞) in E1E1E^(1)\mathcal{E}^{1}E1 we mean that Φ|[s1,s2]Φs1,s2Phi|_([s_(1),s_(2)])\left.\Phi\right|_{\left[s_{1}, s_{2}\right]}Φ|[s1,s2] is a geodesic connecting φ(s1),φ(s2)φs1,φs2varphi(s_(1)),varphi(s_(2))\varphi\left(s_{1}\right), \varphi\left(s_{2}\right)φ(s1),φ(s2) for any s1,s2∈[0,∞)s1,s2∈[0,∞)s_(1),s_(2)in[0,oo)s_{1}, s_{2} \in[0, \infty)s1,s2∈[0,∞) (see (1.12)).
Theorem 3.2 ([60]). Given any test configuration (X,L)(X,L)(X,L)(\mathcal{X}, \mathscr{L})(X,L) for (X,L)(X,L)(X,L)(X, L)(X,L), there exists a geodesic ray Φ(X,L)Φ(X,L)Phi_((X,L))\Phi_{(X, \mathscr{L})}Φ(X,L) emanating from any given smooth potential φ0φ0varphi_(0)\varphi_{0}φ0.
On the other hand, recall from section 2.1 that there is a non-Archimedean potential associated to (X,L)(X,L)(X,L)(\mathcal{X}, \mathscr{L})(X,L) (see (2.1)). Berman-Boucksom-Jonsson proved that there is a direct relation between geodesic rays and non-Archimedean potentials. First they showed that any geodesic ray ΦΦPhi\PhiΦ defines a non-Archimedean potential (cf. (2.1))
ΦNA(v):=−G(v)(Φ), for any v∈XQdivΦNA(v):=−G(v)(Φ), for any v∈XQdivPhi_(NA)(v):=-G(v)(Phi),quad" for any "v inX_(Q)^(div)\Phi_{\mathrm{NA}}(v):=-G(v)(\Phi), \quad \text { for any } v \in X_{\mathbb{Q}}^{\mathrm{div}}ΦNA(v):=−G(v)(Φ), for any v∈XQdiv
where G(v)(Φ)G(v)(Φ)G(v)(Phi)G(v)(\Phi)G(v)(Φ) is the generic Lelong number of ΦΦPhi\PhiΦ considered as a singular quasi-psh potential on a birational model where the center of the valuation G(v)G(v)G(v)G(v)G(v) is a prime divisor.
Theorem 3.3 ([7]). The following statements are true:
(1) The map Φ↦ΦNAΦ↦ΦNAPhi|->Phi_(NA)\Phi \mapsto \Phi_{\mathrm{NA}}Φ↦ΦNA has the image contained in (E1)NAE1NA(E^(1))^(NA)\left(\mathcal{E}^{1}\right)^{\mathrm{NA}}(E1)NA. Conversely, for any ϕ∈ϕ∈phi in\phi \inϕ∈(ε1)NAε1NA(epsi^(1))^(NA)\left(\varepsilon^{1}\right)^{\mathrm{NA}}(ε1)NA, there exists a geodesic ray denoted by γ(ϕ)γ(Ï•)gamma(phi)\gamma(\phi)γ(Ï•) that satisfies γ(ϕ)NA=ϕγ(Ï•)NA=Ï•gamma(phi)_(NA)=phi\gamma(\phi)_{\mathrm{NA}}=\phiγ(Ï•)NA=Ï•.
(2) For any geodesic ray Φ,Φ^=γ(ΦNA)Φ,Φ^=γΦNAPhi, widehat(Phi)=gamma(Phi_(NA))\Phi, \widehat{\Phi}=\gamma\left(\Phi_{\mathrm{NA}}\right)Φ,Φ^=γ(ΦNA) satisfies Φ^NA=ΦNA∈(E1)NAΦ^NA=ΦNA∈E1NAwidehat(Phi)_(NA)=Phi_(NA)in(E^(1))^(NA)\widehat{\Phi}_{\mathrm{NA}}=\Phi_{\mathrm{NA}} \in\left(\mathcal{E}^{1}\right)^{\mathrm{NA}}Φ^NA=ΦNA∈(E1)NA and Φ^≥ΦΦ^≥Φwidehat(Phi) >= Phi\widehat{\Phi} \geq \PhiΦ^≥Φ.
(3) For Φ=γ(ϕ)Φ=γ(Ï•)Phi=gamma(phi)\Phi=\gamma(\phi)Φ=γ(Ï•) with ϕ∈(ε1)NA,E′∞(Φ)=ENA(ϕ)ϕ∈ε1NA,E′∞(Φ)=ENA(Ï•)phi in(epsi^(1))^(NA),E^('oo)(Phi)=E^(NA)(phi)\phi \in\left(\varepsilon^{1}\right)^{\mathrm{NA}}, \mathbf{E}^{\prime \infty}(\Phi)=\mathbf{E}^{\mathrm{NA}}(\phi)ϕ∈(ε1)NA,E′∞(Φ)=ENA(Ï•), and there exists a sequence of test configurations (Xm,Lm)Xm,Lm(X_(m),L_(m))\left(\mathcal{X}_{m}, \mathscr{L}_{m}\right)(Xm,Lm) such that ΦΦPhi\PhiΦ is the decreasing limit of Φ(Xm,Lm)ΦXm,LmPhi_((X_(m),L_(m)))\Phi_{\left(X_{m}, \mathscr{L}_{m}\right)}Φ(Xm,Lm) (see Theorem 3.2).
Berman-Boucksom-Jonsson proved ΦNA∈PSHNAΦNA∈PSHNAPhi_(NA)inPSH^(NA)\Phi_{\mathrm{NA}} \in \mathrm{PSH}^{\mathrm{NA}}ΦNA∈PSHNA by blowing up multiplier ideal sheaves {L(mΦ)}m∈N{L(mΦ)}m∈N{L(m Phi)}_(m inN)\{\mathcal{L}(m \Phi)\}_{m \in \mathbb{N}}{L(mΦ)}m∈N and using their global generation properties to construct test configurations {(Xm,Lm)}Xm,Lm{(X_(m),L_(m))}\left\{\left(\mathcal{X}_{m}, \mathscr{L}_{m}\right)\right\}{(Xm,Lm)} such that ϕ(Xm,Lm)Ï•Xm,Lmphi_((X_(m),L_(m)))\phi_{\left(X_{m}, \mathscr{L}_{m}\right)}Ï•(Xm,Lm) decreases to ΦNAΦNAPhi_(NA)\Phi_{\mathrm{NA}}ΦNA. Because of the second statement, any geodesic ray γ(ϕ)γ(Ï•)gamma(phi)\gamma(\phi)γ(Ï•) with ϕ∈(E1)NAϕ∈E1NAphi in(E^(1))^(NA)\phi \in\left(\mathcal{E}^{1}\right)^{\mathrm{NA}}ϕ∈(E1)NA is called maximal in [7]. By the last statement, maximal geodesic rays can be approximated by (geodesic rays associated to) test configurations. Moreover, when ϕ=ϕ(x,L)∈HNA,γ(ϕ)Ï•=Ï•(x,L)∈HNA,γ(Ï•)phi=phi_((x,L))inH^(NA),gamma(phi)\phi=\phi_{(x, \mathscr{L})} \in \mathscr{H}^{\mathrm{NA}}, \gamma(\phi)Ï•=Ï•(x,L)∈HNA,γ(Ï•) coincides with the geodesic ray from Theorem 3.2. Further useful properties of maximal geodesic rays are known (cf. Theorem 3.1):
Theorem 3.4 ([45]). Let ΦΦPhi\PhiΦ be a maximal geodesic ray.
(1) We have the identity (E−Ric(ω0))′∞(Φ)=(EK)NA(ΦNA)E−Ricâ¡Ï‰0′∞(Φ)=EKNAΦNA(E^(-Ric(omega_(0))))^('oo)(Phi)=(E^(K))^(NA)(Phi_(NA))\left(\mathbf{E}^{-\operatorname{Ric}\left(\omega_{0}\right)}\right)^{\prime \infty}(\Phi)=\left(\mathbf{E}^{K}\right)^{\mathrm{NA}}\left(\Phi_{\mathrm{NA}}\right)(E−Ricâ¡(ω0))′∞(Φ)=(EK)NA(ΦNA).
(2) H′∞(Φ)≥HNA(ΦNA)H′∞(Φ)≥HNAΦNAH^('oo)(Phi) >= H^(NA)(Phi_(NA))\mathbf{H}^{\prime \infty}(\Phi) \geq \mathbf{H}^{\mathrm{NA}}\left(\Phi_{\mathrm{NA}}\right)H′∞(Φ)≥HNA(ΦNA). Moreover, if Φ=Φ(x,L)Φ=Φ(x,L)Phi=Phi_((x,L))\Phi=\Phi_{(x, \mathscr{L})}Φ=Φ(x,L) is associated to a test configuration, then H′∞(Φ)=HNA(ΦNA)H′∞(Φ)=HNAΦNAH^('oo)(Phi)=H^(NA)(Phi_(NA))\mathbf{H}^{\prime \infty}(\Phi)=\mathbf{H}^{\mathrm{NA}}\left(\Phi_{\mathrm{NA}}\right)H′∞(Φ)=HNA(ΦNA).
It is natural to conjecture that H′∞(Φ)=HNA(ΦNA)H′∞(Φ)=HNAΦNAH^('oo)(Phi)=H^(NA)(Phi_(NA))\mathbf{H}^{\prime \infty}(\Phi)=\mathbf{H}^{\mathrm{NA}}\left(\Phi_{\mathrm{NA}}\right)H′∞(Φ)=HNA(ΦNA) always holds for any maximal geodesic ray ΦΦPhi\PhiΦ. This is implied by the algebraic Conjecture 2.12, according to [45, 46].
As pointed out in [7], by a construction of Darvas, there are abundant nonmaximal geodesic rays. In fact, analogous local examples have been used by the author to disprove a conjecture of Demailly on Monge-Ampère mass of psh singularities. It is thus a surprising fact that maximal geodesic rays are the only ones of interest in the cscKcscKcscK\mathrm{cscK}cscK problem.
Theorem 3.5 ([45]). If a geodesic ray ΦΦPhi\PhiΦ satisfies M′∞(Φ)<+∞M′∞(Φ)<+∞M^('oo)(Phi) < +oo\mathbf{M}^{\prime \infty}(\Phi)<+\inftyM′∞(Φ)<+∞, then ΦΦPhi\PhiΦ is maximal.
Note that M′∞(Φ)=lims→+∞M(φ(s))sM′∞(Φ)=lims→+∞ M(φ(s))sM^('oo)(Phi)=lim_(s rarr+oo)(M(varphi(s)))/(s)\mathbf{M}^{\prime \infty}(\Phi)=\lim _{s \rightarrow+\infty} \frac{\mathbf{M}(\varphi(s))}{s}M′∞(Φ)=lims→+∞M(φ(s))s exists by Theorem 1.3. This result resolves a difficulty raised in Boucksom's ICM talk [16], and implies that destabilizing geodesic rays can always be approximated by test configurations, thus giving a very strong evidence for the validity of Yau-Tian-Donaldson Conjecture 3.6. The proof of Theorem 3.5 starts with an equisingularity property ∫X×{|t|<1}e−α(Φ^−Φ)<+∞∫X×{|t|<1} e−α(Φ^−Φ)<+∞int_(X xx{|t| < 1})e^(-alpha( widehat(Phi)-Phi)) < +oo\int_{X \times\{|t|<1\}} e^{-\alpha(\widehat{\Phi}-\Phi)}<+\infty∫X×{|t|<1}e−α(Φ^−Φ)<+∞ for any α>0α>0alpha > 0\alpha>0α>0, and then uses Jensen's inequality, together with a comparison principle, for the EEE\mathbf{E}E functional to get a contradiction with the finite slope assumption if Φ^=γ(ΦNA)≠ΦΦ^=γΦNA≠Φwidehat(Phi)=gamma(Phi_(NA))!=Phi\widehat{\Phi}=\gamma\left(\Phi_{\mathrm{NA}}\right) \neq \PhiΦ^=γ(ΦNA)≠Φ.
3.2. Yau-Tian-Donaldson conjecture for general polarized manifolds
The Yau-Tian-Donaldson (YTD) conjecture says that the existence of canonical Kähler metrics on projective manifolds should be equivalent to a certain K-stability condition. For a general polarization, it is believed that one needs to use a strengthened definition of K-stability such as Definition 2.3. In particular, we have the following version.
Conjecture 3.6 (uniform YTD conjecture). A polarized manifold (X,L)(X,L)(X,L)(X, L)(X,L) admits a cscK metric if and only if (X,L)(X,L)(X,L)(X, L)(X,L) is reduced uniformly KKKKK-stable.
The implication from existence to stability is known, and follows from Theorems 1.4 and 3.1. The other direction is still open in general. However, based on the results discussed thus far, we can explain the proof of a weak version.
Theorem 3.7 ([45]). If (X,L)(X,L)(X,L)(X, L)(X,L) is uniformly stable over models (i.e., there exists γ>0γ>0gamma > 0\gamma>0γ>0 such that MNA(X,L)≥γ⋅JNA(X,L)MNA(X,L)≥γ⋅JNA(X,L)M^(NA)(X,L) >= gamma*J^(NA)(X,L)\mathbf{M}^{\mathrm{NA}}(\mathcal{X}, \mathscr{L}) \geq \gamma \cdot \mathbf{J}^{\mathrm{NA}}(\mathcal{X}, \mathscr{L})MNA(X,L)≥γ⋅JNA(X,L) for any model (X,L))(X,L){:(X,L))\left.(\mathcal{X}, \mathscr{L})\right)(X,L)), then it admits a cscK metric.
Summary of proof. Step 1. By Theorem 1.4, we need to show that MMM\mathbf{M}M is coercive. Assume that the coercivity fails. Then there exists a geodesic ray Φ={φ(s)}s∈[0,∞)Φ={φ(s)}s∈[0,∞)Phi={varphi(s)}_(s in[0,oo))\Phi=\{\varphi(s)\}_{s \in[0, \infty)}Φ={φ(s)}s∈[0,∞) satisfying
M′∞(Φ)≤0,J′∞(Φ)=1,sup(φ(s))=0M′∞(Φ)≤0,J′∞(Φ)=1,sup(φ(s))=0M^('oo)(Phi) <= 0,quadJ^('oo)(Phi)=1,quad s u p(varphi(s))=0\mathbf{M}^{\prime \infty}(\Phi) \leq 0, \quad \mathbf{J}^{\prime \infty}(\Phi)=1, \quad \sup (\varphi(s))=0M′∞(Φ)≤0,J′∞(Φ)=1,sup(φ(s))=0
Such a destabilizing geodesic ray ΦΦPhi\PhiΦ was constructed in [7,27][7,27][7,27][7,27][7,27] from a destabilizing sequence. In this construction, both the convexity of MMM\mathbf{M}M from Theorem 1.3 and a compactness result for potentials with uniform entropy bounds from [6] play crucial roles.
Step 2. By Theorem 3.5, ΦΦPhi\PhiΦ is maximal. Set ϕ=ΦNAÏ•=ΦNAphi=Phi_(NA)\phi=\Phi_{\mathrm{NA}}Ï•=ΦNA. By using Theorem 3.3(3) and Theorem 3.4(1), we derive the identities
Moreover, by Theorem 3.4(2), H′∞(Φ)≥HNA(ϕ)H′∞(Φ)≥HNA(Ï•)H^('oo)(Phi) >= H^(NA)(phi)\mathbf{H}^{\prime \infty}(\Phi) \geq \mathbf{H}^{\mathrm{NA}}(\phi)H′∞(Φ)≥HNA(Ï•) so that M′∞(Φ)≥MNA(ϕ)M′∞(Φ)≥MNA(Ï•)M^('oo)(Phi) >= M^(NA)(phi)\mathbf{M}^{\prime \infty}(\Phi) \geq \mathbf{M}^{\mathrm{NA}}(\phi)M′∞(Φ)≥MNA(Ï•).
Step 3. By Theorem 2.11, there exist models (Xm,Lm)Xm,Lm(X_(m),L_(m))\left(X_{m}, \mathscr{L}_{m}\right)(Xm,Lm) such that ϕm=ϕ(Xm,Lm)Ï•m=Ï•Xm,Lmphi_(m)=phi_((X_(m),L_(m)))\phi_{m}=\phi_{\left(X_{m}, \mathscr{L}_{m}\right)}Ï•m=Ï•(Xm,Lm) converges to ϕÏ•phi\phiÏ• in the strong topology and
There is a version of Theorem 3.7 in [45] when Aut(X,L)Autâ¡(X,L)Aut(X,L)\operatorname{Aut}(X, L)Autâ¡(X,L) is continuous. Moreover, it is shown in [46] that Conjecture 2.12 implies Conjecture 3.6. As mentioned earlier, if ( X,LX,LX,LX, LX,L ) is any polarized spherical manifold, Conjecture 2.12 is true and hence in this case the YTD Conjecture 3.6 is proved. Based on this fact, Delcroix [28] obtained further refined existence results in this case.
We should mention that Sean Paul (see [58]) has works that give a beautiful interpretation of the coercivity of MMM\mathbf{M}M-functional using a new notion of stability for pairs. However, it is not clear how K-stability discussed here can directly imply his stability notion.
3.3. YTD conjecture for Fano varieties
3.3.1. Non-Archimedean approach
Our proof of Theorem 3.7 is in fact modeled on a non-Archimedean approach to the uniform YTD conjecture proposed by Berman-Boucksom-Jonsson in [7]. They carried it out sucessfully for smooth Fano manifolds with discrete automorphism groups. The main advantage in the Fano case is that DNADNAD^(NA)\mathbf{D}^{\mathrm{NA}}DNA satisfies a regularization property and can be used in place of MNAMNAM^(NA)\mathbf{M}^{\mathrm{NA}}MNA to complete the argument. Recently their work has been extended to the most general setting of logloglog\loglog Fano pairs.
Theorem 3.8 ([44, 49,50]). A log Fano pair (X,D)(X,D)(X,D)(X, D)(X,D) admits a Kähler-Einstein metric if and only if it is reduced uniformly stable for all special test configurations.
Note that this combined with Theorem 2.15 also proves the K-polystable version of the YTD conjecture. Theorem 3.8 can be used to get examples of Kähler-Einstein metrics on Fano varieties with large symmetry groups (see, for example, [40]). The proof of Theorem 3.8 is much more technical than [7] because we need to overcome the difficulties caused by singularities. The first key idea is to use an approximation approach initiated in [49]. Consider the log resolution μ:X′→Xμ:X′→Xmu:X^(')rarr X\mu: X^{\prime} \rightarrow Xμ:X′→X as in Section 1.3 and reorganize (1.4) as
where H=μ∗(−KX−D)−∑kθkEkH=μ∗−KX−D−∑k θkEkH=mu^(**)(-K_(X)-D)-sum_(k)theta_(k)E_(k)H=\mu^{*}\left(-K_{X}-D\right)-\sum_{k} \theta_{k} E_{k}H=μ∗(−KX−D)−∑kθkEk is ample by choosing appropriate {θk}θk{theta_(k)}\left\{\theta_{k}\right\}{θk} and Dε=Dε=D_(epsi)=D_{\varepsilon}=Dε=∑k(−ak+ε1+ε)θkEk∑k −ak+ε1+εθkEksum_(k)(-a_(k)+(epsi)/(1+epsi))theta_(k)E_(k)\sum_{k}\left(-a_{k}+\frac{\varepsilon}{1+\varepsilon}\right) \theta_{k} E_{k}∑k(−ak+ε1+ε)θkEk with 0≤ε≪10≤ε≪10 <= epsi≪10 \leq \varepsilon \ll 10≤ε≪1. In [49] we considered the simple case when ak∈ak∈a_(k)ina_{k} \inak∈(−1,0](−1,0](-1,0](-1,0](−1,0] for all kkkkk. In this case for 0<ε≪1,(X′,Dε)0<ε≪1,X′,Dε0 < epsi≪1,(X^('),D_(epsi))0<\varepsilon \ll 1,\left(X^{\prime}, D_{\varepsilon}\right)0<ε≪1,(X′,Dε) is a smooth log Fano pair. A crucial calculation using the valuative criterion from Theorem 2.14 shows that (semi)stability of (X,D)(X,D)(X,D)(X, D)(X,D) implies the uniform stability of (X′,Dε)X′,Dε(X^('),D_(epsi))\left(X^{\prime}, D_{\varepsilon}\right)(X′,Dε) for ε>0ε>0epsi > 0\varepsilon>0ε>0. Moreover, we can prove a version of YTD conjecture for (X′,Dε)X′,Dε(X^('),D_(epsi))\left(X^{\prime}, D_{\varepsilon}\right)(X′,Dε) and deduce that it admits a Kähler-Einstein metric. Next we take a limit as ε→0ε→0epsi rarr0\varepsilon \rightarrow 0ε→0 to get a Kähler-Einstein metric on (X,D)(X,D)(X,D)(X, D)(X,D) itself. The proof of this convergence depends on technical uniform pluripotential and geometric estimates.
In [50], we dealt with the general case when DεDεD_(epsi)D_{\varepsilon}Dε is not necessarily effective. A key difficulty for the argument in [7] to work on singular varieties is that it is not clear how to use multiplier ideal sheaves to approximate a destabilizing geodesic ray ΦΦPhi\PhiΦ when XXXXX is singular. To circumvent this difficulty, we first need to perturb ΦΦPhi\PhiΦ to become a singular quasipsh potential ΦεΦεPhi_(epsi)\Phi_{\varepsilon}Φε on (X′×C,p1′∗Lε)X′×C,p1′∗Lε(X^(')xxC,p_(1)^('**)L_(epsi))\left(X^{\prime} \times \mathbb{C}, p_{1}^{\prime *} L_{\varepsilon}\right)(X′×C,p1′∗Lε). Since X′X′X^(')X^{\prime}X′ is smooth, we know how to approximate ΦεΦεPhi_(epsi)\Phi_{\varepsilon}Φε by test configurations for (X′,Lε)X′,Lε(X^('),L_(epsi))\left(X^{\prime}, L_{\varepsilon}\right)(X′,Lε) thanks to [7]. However, due to the ineffectiveness of DεDεD_(epsi)D_{\varepsilon}Dε, the remaining arguments depend more heavily on non-Archimedean analysis and some key observation on convergence of slopes. In [45] we further derived the valuative criterion for reduced uniform stability and understood how the torus action induces an action on the space of non-Archimedean potentials in order to incorporate group actions in the argument. Note that the non-Archimedean approach a priori does not prove the statement in Theorem 3.8 involving special test configurations. Fortunately, Theorem 2.13 fills this gap.
For completeness, we briefly mention other approaches to the YTD conjecture on Fano manifolds. The classical way to solve the Kähler-Einstein equation is through various continuity methods. Traditionally, one uses Aubin's continuity method involving twisted KE metrics. A more recent continuity method uses KE metrics with edge cone singularities as proposed by Donaldson. Finally, there is a Kähler-Ricci flow approach. Tian's early works showed that the most difficult part in proving the YTD conjecture by continuity methods is to establish the algebraicity of limit objects in the Gromov-Hausdorff topology, and he had essentially reduced this difficulty to proving some partial C0C0C^(0)C^{0}C0-estimates. The partial C0C0C^(0)C^{0}C0 estimates were later proved in different settings, starting with Donaldson-Sun's work in the Kähler-Einstein case, which led to the solution of the YTD conjecture for smooth Fano manifolds in [24,65]. Moreover, the partial C0C0C^(0)C^{0}C0-estimates has applications in constructing moduli spaces of smoothable Kähler-Einstein varieties and proving quasiprojectivity of the moduli spaces of KE manifolds, and these applications preclude the algebraic approach mentioned earlier (see [68]). We also refer to [31,66] for surveys on related topics in this approach.
Very recently, yet another quantization approach is carried out by Kewei Zhang based partly on an earlier work of Rubinstein-Tian-Zhang. Zhang considered an analytic invariant of Moser-Trudinger type, namely
δA(X)=sup{c:supφ∈H∫Xe−c(φ−E(φ))<+∞}δA(X)=supc:supφ∈H ∫X e−c(φ−E(φ))<+∞delta^(A)(X)=s u p{c:s u p_(varphi inH)int_(X)e^(-c(varphi-E(varphi))) < +oo}\delta^{A}(X)=\sup \left\{c: \sup _{\varphi \in \mathscr{H}} \int_{X} e^{-c(\varphi-\mathbf{E}(\varphi))}<+\infty\right\}δA(X)=sup{c:supφ∈H∫Xe−c(φ−E(φ))<+∞}
It is easy to show that the coercivity of DDD\mathbf{D}D-functional is equivalent to δA(X)>1δA(X)>1delta^(A)(X) > 1\delta^{A}(X)>1δA(X)>1. The authors of [61] introduced a quantization δmA(X)δmA(X)delta_(m)^(A)(X)\delta_{m}^{A}(X)δmA(X) by using a quantization of EEE\mathbf{E}E on the space of Bergman metrics, and further proved δmA(X)=δm(X)δmA(X)=δm(X)delta_(m)^(A)(X)=delta_(m)(X)\delta_{m}^{A}(X)=\delta_{m}(X)δmA(X)=δm(X). Using some deep results in complex geometry including Tian's work on Bergman kernels and Berndtsson's subharmonicity theorem, it is proved in [73] that limm→+∞δmA(X)=δA(X)limm→+∞ δmA(X)=δA(X)lim_(m rarr+oo)delta_(m)^(A)(X)=delta^(A)(X)\lim _{m \rightarrow+\infty} \delta_{m}^{A}(X)=\delta^{A}(X)limm→+∞δmA(X)=δA(X). Combining these discussions with the algebraic convergence result of Blum-Jonsson and the valuative criterion of uniform stability of Fujita discussed earlier, Zhang gets δA(X)=δ(X)δA(X)=δ(X)delta^(A)(X)=delta(X)\delta^{A}(X)=\delta(X)δA(X)=δ(X) and completes the proof of uniform version of YTD conjecture for smooth Fano manifolds. It would be interesting to extend this approach to the more general case (i.e., Fano varieties with continuous automorphism groups).
We finish by remarking that it is of interest to apply the ideas and methods from the above two approaches to study the YTD conjecture for general polarizations. For the approach involving partial C0C0C^(0)C^{0}C0-estimates, the geometry is complicated by collapsing phenomenon in the Gromov-Hausdorff convergence with only scalar curvature bounds, which is very difficult to study with current techniques. For the quantization approach, there were some attempts by Mabuchi in several works. But the precise picture seems again unclear.
This work is partially supported by NSF grant (DMS-181086) and an Alfred Sloan Fellowship.
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CHI LI
Department of Mathematics, Rutgers University, 110 Frelinghuysen Rd. Piscataway, NJ 08854-8019, USA, chi.li @ rutgers.edu
THE DOUBLE RAMIFICATION CYCLE FORMULA
AARON PIXTON
ABSTRACT
The double ramification cycle DRg(A)=DRg(μ,ν)DRg(A)=DRg(μ,ν)DR_(g)(A)=DR_(g)(mu,nu)\mathrm{DR}_{g}(A)=\mathrm{DR}_{g}(\mu, \nu)DRg(A)=DRg(μ,ν) is a cycle in the moduli space of stable curves parametrizing genus ggggg curves admitting a map to P1P1P^(1)\mathbb{P}^{1}P1 with specified ramification profiles μ,νμ,νmu,nu\mu, \nuμ,ν over two points. In 2016, Janda, Pandharipande, Zvonkine, and the author proved a formula expressing the double ramification cycle in terms of basic tautological classes, answering a question of Eliashberg from 2001. This formula has an intricate combinatorial shape involving an unusual way to sum divergent series using polynomial interpolation. Here we give some motivation for where this formula came from, relating it both to an older partial formula of Hain and to Givental's R-matrix action on cohomological field theories.
MATHEMATICS SUBJECT CLASSIFICATION 2020
Primary 14N35; Secondary 14H10, 14C17
KEYWORDS
Moduli of curves, double ramification cycle
1. INTRODUCTION
Let g,ng,ng,ng, ng,n be nonnegative integers satisfying 2g−2+n>02g−2+n>02g-2+n > 02 g-2+n>02g−2+n>0, so that the moduli space M¯g,nM¯g,nbar(M)_(g,n)\overline{\mathcal{M}}_{g, n}M¯g,n of stable curves of genus ggggg with nnnnn markings is nonempty. Let A=(a1,…,an)∈ZnA=a1,…,an∈ZnA=(a_(1),dots,a_(n))inZ^(n)A=\left(a_{1}, \ldots, a_{n}\right) \in \mathbb{Z}^{n}A=(a1,…,an)∈Zn be a vector of nnnnn integers satisfying a1+⋯+an=0a1+⋯+an=0a_(1)+cdots+a_(n)=0a_{1}+\cdots+a_{n}=0a1+⋯+an=0. In this paper we will be interested in a Chow cycle class
There are two main perspectives on how to think about and define DRg(A)DRgâ¡(A)DR_(g)(A)\operatorname{DR}_{g}(A)DRgâ¡(A), the double ramification cycle. The first is the source of its name; we think of it as parametrizing the genus ggggg curves CCCCC that admit a finite map C→P1C→P1C rarrP^(1)C \rightarrow \mathbb{P}^{1}C→P1 with specified ramification profiles μ,νμ,νmu,nu\mu, \nuμ,ν over two points (say 0 and ∞∞oo\infty∞ ). These two ramification profiles are encoded in the vector AAAAA : we can take the positive and negative components of AAAAA separately to get two partitions of equal size. The marked points with nonzero aiaia_(i)a_{i}ai should then be the points in CCCCC lying above 0 and ∞∞oo\infty∞, while the marked points with ai=0ai=0a_(i)=0a_{i}=0ai=0 are unconstrained. Ramification above points other than 0 and ∞∞oo\infty∞ is unconstrained.
The above description defines a double ramification locus inside the moduli space of smooth curves Mg,nMg,nM_(g,n)\mathcal{M}_{g, n}Mg,n that is usually (but not always) of pure codimension ggggg. To extend this to a codimension ggggg class on M¯g,nM¯g,nbar(M)_(g,n)\overline{\mathcal{M}}_{g, n}M¯g,n, we can use the virtual class in relative GromovWitten theory. There is a moduli space of stable (rubber) maps to P1P1P^(1)\mathbb{P}^{1}P1 with given marked ramification over two points, M¯g,n(P1/{0,∞},μ,ν)∼M¯g,nP1/{0,∞},μ,ν∼bar(M)_(g,n)(P^(1)//{0,oo},mu,nu)^(∼)\overline{\mathcal{M}}_{g, n}\left(\mathbb{P}^{1} /\{0, \infty\}, \mu, \nu\right)^{\sim}M¯g,n(P1/{0,∞},μ,ν)∼, equipped with a forgetful map ppppp : M¯g,n(P1/{0,∞},μ,ν)∼→M¯g,nM¯g,nP1/{0,∞},μ,ν∼→M¯g,nbar(M)_(g,n)(P^(1)//{0,oo},mu,nu)^(∼)rarr bar(M)_(g,n)\overline{\mathcal{M}}_{g, n}\left(\mathbb{P}^{1} /\{0, \infty\}, \mu, \nu\right)^{\sim} \rightarrow \overline{\mathcal{M}}_{g, n}M¯g,n(P1/{0,∞},μ,ν)∼→M¯g,n, and the double ramification cycle can be taken to be the pushforward under this map of the virtual class,
The second perspective on DRg(A)DRgâ¡(A)DR_(g)(A)\operatorname{DR}_{g}(A)DRgâ¡(A) is via Abel-Jacobi maps. Let Xg→AgXg→AgX_(g)rarrA_(g)\mathcal{X}_{g} \rightarrow \mathcal{A}_{g}Xg→Ag be the universal abelian variety of dimension ggggg. Then the data in the vector AAAAA can be used to define a morphism jA:Mg,n→XgjA:Mg,n→Xgj_(A):M_(g,n)rarrX_(g)j_{A}: \mathcal{M}_{g, n} \rightarrow \mathcal{X}_{g}jA:Mg,n→Xg by
The double ramification locus is then the inverse image under this map of the zero section ZgZgZ_(g)Z_{g}Zg of XgXgX_(g)\mathcal{X}_{g}Xg, since CCCCC admits a map to P1P1P^(1)\mathbb{P}^{1}P1 with the given ramification profiles if and only if OC(a1p1+⋯+anpn)OCa1p1+⋯+anpnO_(C)(a_(1)p_(1)+cdots+a_(n)p_(n))\mathcal{O}_{C}\left(a_{1} p_{1}+\cdots+a_{n} p_{n}\right)OC(a1p1+⋯+anpn) is trivial.
This Abel-Jacobi map extends easily to Mg,nctMg,nctM_(g,n)^(ct)\mathcal{M}_{g, n}^{\mathrm{ct}}Mg,nct, the moduli space of curves of compact type (those with compact Jacobians), but using this perspective to define the double ramification cycle on all of M¯g,nM¯g,nbar(M)_(g,n)\overline{\mathcal{M}}_{g, n}M¯g,n requires more work. It also is not obvious that constructing DRg(A)DRgâ¡(A)DR_(g)(A)\operatorname{DR}_{g}(A)DRgâ¡(A) in this way will give the same class as that given by relative Gromov-Witten theory, even after restriction to Mg,nctMg,nctM_(g,n)^(ct)\mathcal{M}_{g, n}^{\mathrm{ct}}Mg,nct. For one approach to these questions using logarithmic and tropical geometry, see the work of Marcus and Wise [13].
Eliashberg proposed the problem of giving a formula for the double ramification cycle in 2001, in the context of symplectic field theory. This problem was solved by Janda, Pandharipande, Zvonkine, and the author in 2016 [11], giving an explicit combinatorial formula for the double ramification cycle. This formula has an unexpected form-an additional
integer parameter r>0r>0r > 0r>0r>0 is introduced, then an expression is written down that becomes polynomial in rrrrr for rrrrr sufficiently large, and finally this polynomial is specialized to r=0r=0r=0r=0r=0. Subsequent papers extending or generalizing the double ramification cycle formula in various ways (e.g., [2,5,12][2,5,12][2,5,12][2,5,12][2,5,12] ) have left the basic combinatorial structure of the formula virtually unchanged. The purpose of this paper is to discuss this structure and give some motivation for where it comes from.
In Section 2, we review the tautological classes in the Chow ring of the moduli space of stable curves. In Section 3, we discuss results leading up to the formula of [11], most notably Hain's formula for the compact type double ramification cycle. Section 4 is the heart of the paper and consists of an extended discussion motivating the shape of the double ramification cycle formula. We conclude in Section 5 by stating the formula and briefly explaining how its proof in [11] is related to some of the motivation in Section 4.
2. TAUTOLOGICAL CLASSES
2.1. Preliminaries
In this section we review the language in which the double ramification cycle formula is written. This is the language of the tautological ring, a subring R∗(M¯g,n)⊆A∗(M¯g,n)R∗M¯g,n⊆A∗M¯g,nR^(**)( bar(M)_(g,n))subeA^(**)( bar(M)_(g,n))R^{*}\left(\overline{\mathcal{M}}_{g, n}\right) \subseteq A^{*}\left(\overline{\mathcal{M}}_{g, n}\right)R∗(M¯g,n)⊆A∗(M¯g,n) containing most classes that arise naturally in geometry.
Following Faber and Pandharipande [6], the tautological rings R∗(M¯g,n)R∗M¯g,nR^(**)( bar(M)_(g,n))R^{*}\left(\overline{\mathcal{M}}_{g, n}\right)R∗(M¯g,n) can be defined simultaneously for all g,n≥0g,n≥0g,n >= 0g, n \geq 0g,n≥0 satisfying 2g−2+n>02g−2+n>02g-2+n > 02 g-2+n>02g−2+n>0 as the smallest subrings of the Chow rings A∗(M¯g,n)A∗M¯g,nA^(**)( bar(M)_(g,n))A^{*}\left(\overline{\mathcal{M}}_{g, n}\right)A∗(M¯g,n) closed under pushforward by forgetful maps M¯g,n+1→M¯g,nM¯g,n+1→M¯g,nbar(M)_(g,n+1)rarr bar(M)_(g,n)\overline{\mathcal{M}}_{g, n+1} \rightarrow \overline{\mathcal{M}}_{g, n}M¯g,n+1→M¯g,n
and gluing maps M¯g,n+2→M¯g+1,nM¯g,n+2→M¯g+1,nbar(M)_(g,n+2)rarr bar(M)_(g+1,n)\overline{\mathcal{M}}_{g, n+2} \rightarrow \overline{\mathcal{M}}_{g+1, n}M¯g,n+2→M¯g+1,n or M¯g1,n1+1×M¯g2,n2+1→M¯g1+g2,n1+n2M¯g1,n1+1×M¯g2,n2+1→M¯g1+g2,n1+n2bar(M)_(g_(1),n_(1)+1)xx bar(M)_(g_(2),n_(2)+1)rarr bar(M)_(g_(1)+g_(2),n_(1)+n_(2))\overline{\mathcal{M}}_{g_{1}, n_{1}+1} \times \overline{\mathcal{M}}_{g_{2}, n_{2}+1} \rightarrow \overline{\mathcal{M}}_{g_{1}+g_{2}, n_{1}+n_{2}}M¯g1,n1+1×M¯g2,n2+1→M¯g1+g2,n1+n2. Our discussions of tautological classes will use a more explicit description of them. Graber and Pandharipande [8, APPENDIX A] gave a set of additive generators and a multiplication law satisfied by these generators.
These additive generators are formed from three ingredients: psi classes, kappa classes, and generalized gluing maps corresponding to stable graphs. The psi classes ψi∈A1(M¯g,n),i=1,…,nψi∈A1M¯g,n,i=1,…,npsi_(i)inA^(1)( bar(M)_(g,n)),i=1,dots,n\psi_{i} \in A^{1}\left(\overline{\mathcal{M}}_{g, n}\right), i=1, \ldots, nψi∈A1(M¯g,n),i=1,…,n correspond to the nnnnn marked points and are defined as the first Chern classes of the cotangent line bundles to the curves at those points. The ArbarelloCornalba [1] kappa classes are then the pushforwards of powers of psi classes,
where π:M¯g,n+1→M¯g,nÏ€:M¯g,n+1→M¯g,npi: bar(M)_(g,n+1)rarr bar(M)_(g,n)\pi: \overline{\mathcal{M}}_{g, n+1} \rightarrow \overline{\mathcal{M}}_{g, n}Ï€:M¯g,n+1→M¯g,n forgets the last marking. The kappa classes will not appear in any of the formulas in this paper.
The tautological ring of the moduli space of smooth marked curves, R∗(Mg,n)R∗Mg,nR^(**)(M_(g,n))R^{*}\left(\mathcal{M}_{g, n}\right)R∗(Mg,n), is the ring generated by these ψiψipsi_(i)\psi_{i}ψi and κaκakappa_(a)\kappa_{a}κa. To extend this to R∗(M¯g,n)R∗M¯g,nR^(**)( bar(M)_(g,n))R^{*}\left(\overline{\mathcal{M}}_{g, n}\right)R∗(M¯g,n) we need classes supported on boundary strata.
2.2. Stable graphs
A stable graph ΓΓGamma\GammaΓ is the combinatorial data of a boundary stratum in M¯g,nM¯g,nbar(M)_(g,n)\overline{\mathcal{M}}_{g, n}M¯g,n. It consists of the following:
(1) a set of vertices V(Γ)V(Γ)V(Gamma)V(\Gamma)V(Γ);
(2) a genus gv≥0gv≥0g_(v) >= 0g_{v} \geq 0gv≥0 at each vertex v∈V(Γ)v∈V(Γ)v in V(Gamma)v \in V(\Gamma)v∈V(Γ);
(3) a set of half-edges H(Γ)H(Γ)H(Gamma)H(\Gamma)H(Γ);
(4) an incidence map H(Γ)→V(Γ)H(Γ)→V(Γ)H(Gamma)rarr V(Gamma)H(\Gamma) \rightarrow V(\Gamma)H(Γ)→V(Γ);
(5) a partition of H(Γ)H(Γ)H(Gamma)H(\Gamma)H(Γ) into sets of size 2 (called edges, the set of which is denoted E(Γ))E(Γ))E(Gamma))E(\Gamma))E(Γ)) and sets of size 1 (called legs);
(6) a bijection between the set of legs and {1,…,n}{1,…,n}{1,dots,n}\{1, \ldots, n\}{1,…,n}.
The underlying graph is required to be connected. The stability constraint is that
at each vertex vvvvv, where nvnvn_(v)n_{v}nv is the number of half-edges incident to vvvvv. The genera are constrained by the identity
2g−2+n=∑v∈V(Γ)(2gv−2+nv)2g−2+n=∑v∈V(Γ) 2gv−2+nv2g-2+n=sum_(v in V(Gamma))(2g_(v)-2+n_(v))2 g-2+n=\sum_{v \in V(\Gamma)}\left(2 g_{v}-2+n_{v}\right)2g−2+n=∑v∈V(Γ)(2gv−2+nv)
or equivalently that g−∑vgv=h1(Γ)g−∑v gv=h1(Γ)g-sum_(v)g_(v)=h^(1)(Gamma)g-\sum_{v} g_{v}=h^{1}(\Gamma)g−∑vgv=h1(Γ), the first Betti number of the graph. Such a stable graph ΓΓGamma\GammaΓ corresponds to a generalized gluing map
where ΓΓGamma\GammaΓ is a stable graph and ααalpha\alphaα is a monomial in the psi and kappa classes on the M¯gv,nvM¯gv,nvbar(M)_(g_(v),n_(v))\overline{\mathcal{M}}_{g_{v}, n_{v}}M¯gv,nv factors. These are the additive generators for the tautological ring considered in [8].
2.3. Compact type
The moduli space of curves of compact type, denoted Mg,nctMg,nctM_(g,n)^(ct)\mathcal{M}_{g, n}^{\mathrm{ct}}Mg,nct, is the open subscheme of M¯g,nM¯g,nbar(M)_(g,n)\overline{\mathcal{M}}_{g, n}M¯g,n consisting of those curves whose dual graph is a tree. Its tautological ring R∗(Mg,nct)R∗Mg,nctR^(**)(M_(g,n)^(ct))R^{*}\left(\mathcal{M}_{g, n}^{\mathrm{ct}}\right)R∗(Mg,nct) is the image of R∗(M¯g,n)R∗M¯g,nR^(**)( bar(M)_(g,n))R^{*}\left(\overline{\mathcal{M}}_{g, n}\right)R∗(M¯g,n) under restriction, so it is additively generated by classes ξΓ∗(α)ξΓ∗(α)xi_(Gamma_(**))(alpha)\xi_{\Gamma_{*}}(\alpha)ξΓ∗(α) as above where ΓΓGamma\GammaΓ is a tree.
It will be convenient for us to have notation for the compact type boundary divisor classes when stating Hain's formula below, (3.2). If ΓΓGamma\GammaΓ is a stable graph with 2 vertices and 1 edge and one of the vertices is genus hhhhh and has those legs with markings in a set S⊆1,2,…,nS⊆1,2,…,nS sube1,2,dots,nS \subseteq 1,2, \ldots, nS⊆1,2,…,n, let δh,S=ξΓ∗(1)δh,S=ξΓ∗(1)delta_(h,S)=xi_(Gamma_(**))(1)\delta_{h, S}=\xi_{\Gamma_{*}}(1)δh,S=ξΓ∗(1) be the corresponding boundary divisor class.
3. PREVIOUS FORMULAS AND RESULTS
The first progress towards a formula for the double ramification cycle was when Faber and Pandharipande [7] proved that the double ramification cycle lies in the tautological ring, and thus in theory must be expressible in terms of the generators described in
the previous section. Their proof, although in principle constructive, involves a complicated recursion and does not seem to yield a practical formula.
The first progress towards an explicit formula came when Hain [10] computed the double ramification cycle when restricted to the compact type locus Mg,nctMg,nctM_(g,n)^(ct)\mathcal{M}_{g, n}^{\mathrm{ct}}Mg,nct. On this locus the double ramification cycle is the pullback along an Abel-Jacobi map jA:Mg,nct→Xg,njA:Mg,nct→Xg,nj_(A):M_(g,n)^(ct)rarrX_(g,n)j_{A}: \mathcal{M}_{g, n}^{\mathrm{ct}} \rightarrow \mathcal{X}_{g, n}jA:Mg,nct→Xg,n of the class of the zero section Zg,nZg,nZ_(g,n)\mathcal{Z}_{g, n}Zg,n of the universal abelian variety Xg,n→Ag,nXg,n→Ag,nX_(g,n)rarrA_(g,n)\mathcal{X}_{g, n} \rightarrow \mathcal{A}_{g, n}Xg,n→Ag,n. Hain showed that the class of this zero section is [Zg,n]=Θg/gZg,n=Θg/g[Z_(g,n)]=Theta^(g)//g\left[\mathcal{Z}_{g, n}\right]=\Theta^{g} / g[Zg,n]=Θg/g ! and computed the pullback of the theta divisor ΘΘTheta\ThetaΘ as an explicit divisor on Mg,nctMg,nctM_(g,n)^(ct)\mathcal{M}_{g, n}^{\mathrm{ct}}Mg,nct,
where aS=∑i∈SaiaS=∑i∈S aia_(S)=sum_(i in S)a_(i)a_{S}=\sum_{i \in S} a_{i}aS=∑i∈Sai and the second sum runs over all h,Sh,Sh,Sh, Sh,S defining boundary divisor classes.
Hain's formula for the compact type double ramification cycle is then
The divisor formula (3.1) is a homogeneous polynomial of degree 2 in AAAAA, so Hain's DR formula (3.2) is a homogeneous polynomial of degree 2g2g2g2 g2g in AAAAA.
Grushevsky and Zakharov [9] extended Hain's computation slightly, expanding from Mg,nctMg,nctM_(g,n)^(ct)\mathcal{M}_{g, n}^{\mathrm{ct}}Mg,nct to a slightly larger open subscheme of M¯g,nM¯g,nbar(M)_(g,n)\overline{\mathcal{M}}_{g, n}M¯g,n by adding the locus of curves whose dual graph is a tree with a single loop added at one vertex. If ΓΓGamma\GammaΓ is the stable graph with a single vertex and single loop, then their correction term is the codimension ggggg part of
where ψ1,…,ψnψ1,…,ψnpsi_(1),dots,psi_(n)\psi_{1}, \ldots, \psi_{n}ψ1,…,ψn are the psi classes on the legs, ψ,ψ′ψ,ψ′psi,psi^(')\psi, \psi^{\prime}ψ,ψ′ are the psi classes on the two halfedges of the loop, and B2kB2kB_(2k)B_{2 k}B2k is a Bernoulli number.
In particular, the double ramification cycle is no longer a homogeneous polynomial in AAAAA when computed beyond compact type. This was also seen in work of Buryak, Shadrin, Spitz, and Zvonkine [3], who showed that the top degree intersections of double ramification cycles with monomials in the psi classes are inhomogeneous polynomials of degree 2g2g2g2 g2g in AAAAA.
4. MOTIVATION FOR THE FORMULA
In this section we discuss various observations and ideas that come about when
one tries to extend Hain's formula (3.2) to M¯g,nM¯g,nbar(M)_(g,n)\overline{\mathcal{M}}_{g, n}M¯g,n to obtain a full double ramification cycle formula.
4.1. Expanding Hain's formula
Exponentiating a boundary divisor class can be done using the multiplication laws for tautological classes [8, APPENDIX A]. Multiplying out Hain's formula (3.2) in this way gives
a nice sum over trees: DRgct(A)DRgctâ¡(A)DR_(g)^(ct)(A)\operatorname{DR}_{g}^{c t}(A)DRgctâ¡(A) is the codimension ggggg part of
where the function w:H(T)→Zw:H(T)→Zw:H(T)rarrZw: H(T) \rightarrow \mathbb{Z}w:H(T)→Z is defined here by contracting all the edges in the tree TTTTT other than the one containing hhhhh and then letting w(h)w(h)w(h)w(h)w(h) be the sum of the aiaia_(i)a_{i}ai for the legs iiiii on the same vertex as the half-edge hhhhh.
Extending this formula to M¯g,nM¯g,nbar(M)_(g,n)\overline{\mathcal{M}}_{g, n}M¯g,n requires us to provide a polynomial (or power series) in the psi classes for every stable graph ΓΓGamma\GammaΓ, not just every stable tree. The w(h)w(h)w(h)w(h)w(h) definition above does not naturally extend to non-separating edges, so it is not immediately clear how to do this. Moreover, we know that this power series needs to be (3.3) for the single-loop graph, so something quite new is going on even there.
j∗DRg(a1,…,an)=DRg1({ai∣i∈I1},t)⊗DRg2({ai∣i∈I2},−t)j∗DRgâ¡a1,…,an=DRg1â¡ai∣i∈I1,t⊗DRg2â¡ai∣i∈I2,−tj^(**)DR_(g)(a_(1),dots,a_(n))=DR_(g_(1))({a_(i)∣i inI_(1)},t)oxDR_(g_(2))({a_(i)∣i inI_(2)},-t)j^{*} \operatorname{DR}_{g}\left(a_{1}, \ldots, a_{n}\right)=\operatorname{DR}_{g_{1}}\left(\left\{a_{i} \mid i \in I_{1}\right\}, t\right) \otimes \operatorname{DR}_{g_{2}}\left(\left\{a_{i} \mid i \in I_{2}\right\},-t\right)j∗DRgâ¡(a1,…,an)=DRg1â¡({ai∣i∈I1},t)⊗DRg2â¡({ai∣i∈I2},−t)
where t∈Zt∈Zt inZt \in \mathbb{Z}t∈Z is the unique insertion that makes the parameters sum to 0 in each DR term on the right.
If the double ramification cycle were a CohFT, we would want a similar formula for the pullback along the nonseparating gluing map k:M¯g−1,n+2→M¯g,nk:M¯g−1,n+2→M¯g,nk: bar(M)_(g-1,n+2)rarr bar(M)_(g,n)k: \overline{\mathcal{M}}_{g-1, n+2} \rightarrow \overline{\mathcal{M}}_{g, n}k:M¯g−1,n+2→M¯g,n : the natural thing to write down would be
but it is not clear how one might make sense of this infinite sum-it will not converge in any standard sense. What is going wrong here is that CohFTs are supposed to depend multilinearly on parameters from a finite-dimensional state space, but double ramification cycles take inputs in ZZZ\mathbb{Z}Z so the state space appears to be infinite-dimensional.
So the double ramification cycle behaves like a CohFT as far as separating nodes are concerned, but the state space would have to be infinite-dimensional and this makes it unclear what to do at nonseparating nodes.
4.3. Givental's R-matrix action
Teleman [16] proved that semisimple CohFTs all have a very particular graph sum form, given by applying Givental's R-matrix action to a CohFT that lives fully in codimension zero. The rough shape of the resulting formula for a semisimple CohFT is
where the second sum is over functions wwwww on the half-edges of the graph taking values in some set SSSSS (a basis for the state space of the CohFT) and the values of wwwww on the legs h1,…,hnh1,…,hnh_(1),dots,h_(n)h_{1}, \ldots, h_{n}h1,…,hn are given as w(hi)=γiwhi=γiw(h_(i))=gamma_(i)w\left(h_{i}\right)=\gamma_{i}w(hi)=γi. The various factors are then power series (that depend on wwwww ) in the corresponding kappa and psi classes. The expanded version of Hain's compact type formula (4.1) is of this shape: we take S=ZS=ZS=ZS=\mathbb{Z}S=Z, the vertex factor is 0 unless all of the incident w(h)w(h)w(h)w(h)w(h) sum to zero, and the edge factor is 0 unless the two w(h)w(h)w(h)w(h)w(h) along the edge sum to zero. These vanishings effectively place the following constraints on wwwww (to get a nonzero contribution to DRgct(A))DRgct(A){:DR_(g)^(ct)(A))\left.\mathrm{DR}_{g}^{\mathrm{ct}}(A)\right)DRgct(A)) :
(1) w(hi)=aiwhi=aiw(h_(i))=a_(i)w\left(h_{i}\right)=a_{i}w(hi)=ai for i=1,2,…,ni=1,2,…,ni=1,2,dots,ni=1,2, \ldots, ni=1,2,…,n, where hihih_(i)h_{i}hi is the iiiii th leg;
(2) w(h)+w(h′)=0w(h)+wh′=0w(h)+w(h^('))=0w(h)+w\left(h^{\prime}\right)=0w(h)+w(h′)=0 if {h,h′}h,h′{h,h^(')}\left\{h, h^{\prime}\right\}{h,h′} is an edge;
(3) ∑h→vw(h)=0∑h→v w(h)=0sum_(h rarr v)w(h)=0\sum_{h \rightarrow v} w(h)=0∑h→vw(h)=0 for each vertex vvvvv.
We say wwwww is balanced (with respect to AAAAA ) if it satisfies these constraints. In other words, wwwww is a flow on ΓΓGamma\GammaΓ with sources/sinks at the legs (with specified values given there by AAAAA ). When ΓΓGamma\GammaΓ is a tree, there is a unique such balanced wwwww and we recover the w(h)w(h)w(h)w(h)w(h) used in (4.1).
From this perspective it is natural to just try to take (4.1) and extend it to be a Givental-type sum over arbitrary graphs (not just trees), but then there will be infinitely many choices of wwwww and the resulting infinite sums will be nonconvergent. Moreover, careful comparison with the exact form of Givental's R-matrix action suggests that the vertex factor should contribute a total factor of something like " |Z|−h1(Γ)|Z|−h1(Γ)|Z|^(-h^(1)(Gamma))|\mathbb{Z}|^{-h^{1}(\Gamma)}|Z|−h1(Γ)." Note that the set of balanced wwwww is a torsor over H1(Γ;Z)≅Zh1(Γ)H1(Γ;Z)≅Zh1(Γ)H_(1)(Gamma;Z)~=Z^(h^(1)(Gamma))H_{1}(\Gamma ; \mathbb{Z}) \cong \mathbb{Z}^{h^{1}(\Gamma)}H1(Γ;Z)≅Zh1(Γ), so this factor feels like some sort of infinite averaging procedure.
4.4. Divergent averages
Returning to the simplest non-tree case, the graph with one vertex and one loop, matching things up with (3.3) would then require making sense of the "infinite average" identity
but there is no obvious way to make sense of this similarity. Moreover, more complicated graphs require much more complicated divergent sums; for example, a graph with two vertices, a double edge between them, and one leg on each vertex gives rise to infinite sums like
The problem with writing down a double ramification cycle formula of this type is clearly that the state space is infinite-dimensional. If we replace ZZZ\mathbb{Z}Z with Z/rZZ/rZZ//rZ\mathbb{Z} / r \mathbb{Z}Z/rZ everywhere then there is no difficulty with writing down a similar-looking finite rank CohFT. The result might be something like the following (the case of a diagonal R-matrix-for an example of a more complicated CohFT of this general type, see [15]):
for power series Fa(Z)Fa(Z)F_(a)(Z)F_{a}(Z)Fa(Z) for a∈Z/rZa∈Z/rZa inZ//rZa \in \mathbb{Z} / r \mathbb{Z}a∈Z/rZ with F0(Z)=0F0(Z)=0F_(0)(Z)=0F_{0}(Z)=0F0(Z)=0 and F−a(−Z)=−Fa(Z)F−a(−Z)=−Fa(Z)F_(-a)(-Z)=-F_(a)(Z)F_{-a}(-Z)=-F_{a}(Z)F−a(−Z)=−Fa(Z).
If we take Fa(Z)=12a2ZFa(Z)=12a2ZF_(a)(Z)=(1)/(2)a^(2)ZF_{a}(Z)=\frac{1}{2} a^{2} ZFa(Z)=12a2Z for −r2<a≤r2−r2<a≤r2-(r)/(2) < a <= (r)/(2)-\frac{r}{2}<a \leq \frac{r}{2}−r2<a≤r2 then this CohFT starts to look very much like the expanded version of Hain's formula, (4.1). In fact, if ΓΓGamma\GammaΓ is a tree then the ΓΓGamma\GammaΓ-term in this sum agrees with that in Hain's formula for all sufficiently large rrrrr. So it is tempting to try to take the limit as r→∞r→∞r rarr oor \rightarrow \inftyr→∞ of these CohFTs. But this is not quite right: the rrrrr-version of the left side of (4.2) is then
1r∑−r2<c≤r2c2k1r∑−r2<c≤r2 c2k(1)/(r)sum_(-(r)/(2) < c <= (r)/(2))c^(2k)\frac{1}{r} \sum_{-\frac{r}{2}<c \leq \frac{r}{2}} c^{2 k}1r∑−r2<c≤r2c2k
This certainly does not converge as r→∞r→∞r rarr oor \rightarrow \inftyr→∞. However, if we restrict to even rrrrr then it is polynomial in rrrrr, and if we examine the coefficients of this polynomial then we see that B2kB2kB_(2k)B_{2 k}B2k, the desired value, is the constant coefficient in rrrrr.
This suggests a potential interpretation even of more complicated sums like (4.3):
where ccccc and ddddd must be interpreted inside c2kd2lc2kd2lc^(2k)d^(2l)c^{2 k} d^{2 l}c2kd2l as elements of ZZZ\mathbb{Z}Z via some choice of modrmodrmod r\bmod rmodr representatives (we used −r/2+1,…,r/2−r/2+1,…,r/2-r//2+1,dots,r//2-r / 2+1, \ldots, r / 2−r/2+1,…,r/2 before but 0,…,r−10,…,r−10,dots,r-10, \ldots, r-10,…,r−1 will give the same final answer) and setting r=0r=0r=0r=0r=0 at the end is done by polynomial interpolation.
4.6. Geometric interpretation from (k/r)(k/r)(k//r)(k / r)(k/r)-spin structures
An (k/r)(k/r)(k//r)(k / r)(k/r)-spin structure on a smooth curve CCCCC with marked points pipip_(i)p_{i}pi and weights aiaia_(i)a_{i}ai is a choice of line bundle LLLLL on CCCCC such that L⊗r≡ωC⊗k(a1p1+⋯+anpn)L⊗r≡ωC⊗ka1p1+⋯+anpnL^(ox r)-=omega_(C)^(ox k)(a_(1)p_(1)+cdots+a_(n)p_(n))L^{\otimes r} \equiv \omega_{C}^{\otimes k}\left(a_{1} p_{1}+\cdots+a_{n} p_{n}\right)L⊗r≡ωC⊗k(a1p1+⋯+anpn). If we take k=0k=0k=0k=0k=0 and assume the weights aiaia_(i)a_{i}ai sum to zero then for any positive rrrrr any smooth curve will have such " rrrrr th root structures." But we can also interpret this construction as meaningful when r=0r=0r=0r=0r=0, when we get that a curve only admits a (0/0)-spin structure if it is in the double ramification locus. This observation gives a vague geometric idea for what it might mean to think of the double ramification cycle as given by specializing some parameter rrrrr to 0 .
5. THE DOUBLE RAMIFICATION CYCLE FORMULA
We can now state the main result of [11], the double ramification cycle formula:
Theorem 1 ([11]). DRg(A)DRg(A)DR_(g)(A)\mathrm{DR}_{g}(A)DRg(A) is the codimension ggggg part of
(for PPPPP a polynomial) are evaluated by setting r=0r=0r=0r=0r=0 in the corresponding rrrrr-polynomial
1rh1(Γ)∑w:H(Γ)→{0,1,…,r−1} balanced mod rP({w(h)})1rh1(Γ)∑w:H(Γ)→{0,1,…,r−1} balanced mod r P({w(h)})(1)/(r^(h^(1)(Gamma)))sum_({:[w:H(Gamma)rarr{0","1","dots","r-1}],[" balanced mod "r]:})P({w(h)})\frac{1}{r^{h^{1}(\Gamma)}} \sum_{\substack{w: H(\Gamma) \rightarrow\{0,1, \ldots, r-1\} \\ \text { balanced mod } r}} P(\{w(h)\})1rh1(Γ)∑w:H(Γ)→{0,1,…,r−1} balanced mod rP({w(h)})
The combinatorial result (necessary for this theorem statement to make sense) that the expression in the final line is in fact a polynomial in rrrrr (for rrrrr sufficiently large) was proved in [11, APPENDIX A].
The proof of Theorem 1 in [11] follows some of the motivation in Section 4. We first explain the meaning of the additional rrrrr parameter. For each r>0r>0r > 0r>0r>0, let P1[r]P1[r]P^(1)[r]\mathbb{P}^{1}[r]P1[r] denote the projective line with a BZrBZrBZ_(r)B \mathbb{Z}_{r}BZr orbifold point at 0 . One can then use C∗C∗C^(**)\mathbb{C}^{*}C∗-localization on the moduli space of relative stable maps to P1[r]/{∞}P1[r]/{∞}P^(1)[r]//{oo}\mathbb{P}^{1}[r] /\{\infty\}P1[r]/{∞} to obtain complicated relations that entangle double ramification cycles, classes coming from moduli of (0/r)(0/r)(0//r)(0 / r)(0/r)-spin curves (discussed briefly in the case of smooth curves in Section 4.6), and other basic tautological classes. The relevant (0/r)(0/r)(0//r)(0 / r)(0/r)-spin classes were previously computed by Chiodo [4] using GrothendieckRiemann-Roch.
These localization relations are too difficult to study effectively for specific values of rrrrr, but it turns out that they have polynomial dependence on rrrrr. Taking the constant term in rrrrr simplifies them greatly: most of the terms vanish, and the only remaining terms are a
single double ramification cycle and the r=0r=0r=0r=0r=0 interpolation of certain classes written in terms of the Chern characters of the pushforward of the universal rrrrr th root line bundle on the moduli space of (0/r)(0/r)(0//r)(0 / r)(0/r)-spin curves. Chiodo's formula [4] for these Chern characters gives that these classes are CohFTs with formulas of the type described in Section 4.5. The power series in psi classes appearing in these formulas do not look exactly like those appearing in Theorem 1, but they have the same r=0r=0r=0r=0r=0 interpolation. (In the language of Section 4.5, the power series Fa(Z)Fa(Z)F_(a)(Z)F_{a}(Z)Fa(Z) will be congruent to 12a2Zmodr12a2Zmodr(1)/(2)a^(2)Z mod r\frac{1}{2} a^{2} Z \bmod r12a2Zmodr.) The result is a proof of Theorem 1 .
ACKNOWLEDGMENTS
The author is grateful to R. Cavalieri, D. Holmes, F. Janda, S. Grushevsky, S. Molcho, G. Oberdieck, R. Pandharipande, J. Schmitt, D. Zakharov, and D. Zvonkine for many discussions about double ramification cycles and related topics over the years.
FUNDING
The author was partially supported by National Science Foundation Grant No. 1807079.
REFERENCES
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[4] A. Chiodo, Towards an enumerative geometry of the moduli space of twisted curves and rrrrr th roots. Compos. Math. 144 (2008), no. 6, 1461-1496.
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[6] C. Faber and R. Pandharipande, Logarithmic series and Hodge integrals in the tautological ring. Michigan Math. J. 48 (2000), 215-252.
[7] C. Faber and R. Pandharipande, Relative maps and tautological classes. J. Eur. Math. Soc. (JEMS) 7 (2005), no. 1, 13-49.
[8] T. Graber and R. Pandharipande, Constructions of nontautological classes on moduli spaces of curves. Michigan Math. J. 51 (2003), no. 1, 93-109.
[9] S. Grushevsky and D. Zakharov, The zero section of the universal semiabelian variety and the double ramification cycle. Duke Math. J. 163 (2014), no. 5, 953-982.
[10] R. Hain, Normal functions and the geometry of moduli spaces of curves. In Handbook of moduli. Vol. I, pp. 527-578, Adv. Lect. Math. (ALM) 24, Int. Press, Somerville, MA, 2013.
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[12] F. Janda, R. Pandharipande, A. Pixton, and D. Zvonkine, Double ramification cycles with target varieties. J. Topol. 13 (2020), no. 4, 1725-1766.
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[15] R. Pandharipande, A. Pixton, and D. Zvonkine, Tautological relations via rrrrr-spin structures. J. Algebraic Geom. 28 (2019), no. 3, 439-496.
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AARON PIXTON
Department of Mathematics, University of Michigan, Ann Arbor, MI 48109, USA, pixton@umich.edu
EFFECTIVE RESULTS IN THE THREE-DIMENSIONAL MINIMAL MODEL PROGRAM
YURI PROKHOROV
ABSTRACT
We give a brief review on recent developments in the three-dimensional minimal model program.
MATHEMATICS SUBJECT CLASSIFICATION 2020
Primary 14E30; Secondary 14J30, 14B05, 14E05
KEYWORDS
Minimal model, Mori contraction, terminal (canonical) singularity, flip, extremal ray
In this note we give a brief review on recent developments in the three-dimensional minimal model program (MMP for short). Certainly, this is not a complete survey of all advances in this area. For example, we do not discuss the minimal models of varieties of nonnegative Kodaira dimension, as well as applications to birational geometry and moduli spaces.
The aim of the MMP is to find a good representative in a fixed birational equivalence class of algebraic varieties. Starting with an arbitrary smooth projective variety, one can perform a finite number of certain elementary transformations, called divisorial contractions and flips, and at the end obtain a variety which is simpler in some sense. Most parts of the MMP are completed in arbitrary dimension. One of the basic remaining problems is the following:
Describe all the intermediate steps and the outcome of the MMP.
The MMP makes sense only in dimensions ≥2≥2>= 2\geq 2≥2, and for surfaces it is classical and well known. So the first nontrivial case is the three-dimensional one. It turns out that to proceed with the MMP in dimension ≥3≥3>= 3\geq 3≥3, one has to work with varieties admitting certain types of very mild, the so-called terminal, singularities. On the other hand, dimension 3 is the last dimension where one can expect effective results: in higher dimensions, classification results become very complicated and unreasonably long.
We will work over the field CCC\mathbb{C}C of complex numbers throughout. A variety is either an algebraic variety or a reduced complex space.
1. SINGULARITIES
Recall that a Weil divisor DDDDD on a normal variety is said to be QQQ\mathbb{Q}Q-Cartier if its multiple nDnDnDn DnD, for some nnnnn, is a Cartier divisor. For any morphism f:Y→Xf:Y→Xf:Y rarr Xf: Y \rightarrow Xf:Y→X, the pull-back f∗Df∗Df^(**)Df^{*} Df∗D of a QQQ\mathbb{Q}Q-Cartier divisor DDDDD is well defined as a divisor with rational coefficients ( QQQ\mathbb{Q}Q-divisor). A variety XXXXX has QQQ\mathbb{Q}Q-factorial singularities if any Weil divisor on XXXXX is QQQ\mathbb{Q}Q-Cartier.
Definition 1.1. A normal algebraic variety (or an analytic space) XXXXX is said to have terminal (resp. canonical, log terminal, log canonical) singularities if the canonical Weil divisor KXKXK_(X)K_{X}KX is QQQ\mathbb{Q}Q-Cartier and, for any birational morphism f:Y→Xf:Y→Xf:Y rarr Xf: Y \rightarrow Xf:Y→X, one can write
where EiEiE_(i)E_{i}Ei are all the exceptional divisors and ai>0ai>0a_(i) > 0a_{i}>0ai>0 (resp. ai≥0,ai>−1,ai≥−1ai≥0,ai>−1,ai≥−1a_(i) >= 0,a_(i) > -1,a_(i) >= -1a_{i} \geq 0, a_{i}>-1, a_{i} \geq-1ai≥0,ai>−1,ai≥−1 ) for all iiiii. The smallest positive mmmmm such that mKXmKXmK_(X)m K_{X}mKX is Cartier is called the Gorenstein index of XXXXX. Canonical singularities of index 1 are rational Gorenstein.
The class of terminal QQQ\mathbb{Q}Q-factorial singularities is the smallest class that is closed under the MMP. Canonical singularities are important because they appear in the canonical models of varieties of general type. A crucial observation is that terminal singularities lie in codimension ≥3≥3>= 3\geq 3≥3. In particular, terminal surface singularities are smooth and terminal threefold singularities are isolated. Canonical singularities of surfaces are called DuValDuValDuVal\mathrm{Du} \mathrm{Val}DuVal
or rational double points. Any two-dimensional log terminal singularity is a quotient of a smooth surface germ by a finite group [32]. Terminal threefolds singularities were classified by M. Reid [65] and S. Mori [43].
Example. Let X⊂C4X⊂C4X subC^(4)X \subset \mathbb{C}^{4}X⊂C4 be a hypersurface given by the equation
where ϕ=0Ï•=0phi=0\phi=0Ï•=0 is an equation of a DuValDuValDuVal\mathrm{Du} \operatorname{Val}DuVal (ADE) singularity. Then the singularity of XXXXX at 0 is canonical Gorenstein. It is terminal if and only if it is isolated. Singularities of this type are called cDV.
According to [65], any three-dimensional terminal singularity of index m>1m>1m > 1m>1m>1 is a quotient of an isolated cDV-singularity by the cyclic group μmμmmu_(m)\boldsymbol{\mu}_{m}μm of order mmmmm. More precisely, we have the following
Theorem 1.2 ([65]). Let (X∋P)(X∋P)(X∋P)(X \ni P)(X∋P) be an analytic germ of a three-dimensional terminal singularity of index m≥1m≥1m >= 1m \geq 1m≥1. Then there exist an isolated cDVcDVcDV\mathrm{cDV}cDV-singularity (X♯∋P♯)X♯∋P♯(X^(♯)∋P^(♯))\left(X^{\sharp} \ni P^{\sharp}\right)(X♯∋P♯) and a cyclic μmμmmu_(m)\mu_{m}μm-cover
The morphism πÏ€pi\piÏ€ in the above theorem is called the index-one cover. A detailed classification of all possibilities for the equations of X♯⊂C4X♯⊂C4X^(♯)subC^(4)X^{\sharp} \subset \mathbb{C}^{4}X♯⊂C4 and the actions of μmμmmu_(m)\boldsymbol{\mu}_{m}μm was obtained in [43] (see also [66]).
Example. Let the cyclic group μmμmmu_(m)\mu_{m}μm act on CnCnC^(n)\mathbb{C}^{n}Cn diagonally via
Then we say that (a1,…,an)a1,…,an(a_(1),dots,a_(n))\left(a_{1}, \ldots, a_{n}\right)(a1,…,an) is the collection of weights of the action. Assume that the action is free in codimension 1 . Then the quotient singularity Cn/μm∋0Cn/μm∋0C^(n)//mu_(m)∋0\mathbb{C}^{n} / \mu_{m} \ni 0Cn/μm∋0 is said to be of type 1m(a1,…,an)1ma1,…,an(1)/(m)(a_(1),dots,a_(n))\frac{1}{m}\left(a_{1}, \ldots, a_{n}\right)1m(a1,…,an). According to the criterion (see [66, THEOREM 4.11]), this singularity is terminal if and only if
∑i=1nkai¯>m for k=1,…,m−1∑i=1n kai¯>m for k=1,…,m−1sum_(i=1)^(n) bar(ka_(i)) > m quad" for "k=1,dots,m-1\sum_{i=1}^{n} \overline{k a_{i}}>m \quad \text { for } k=1, \ldots, m-1∑i=1nkai¯>m for k=1,…,m−1
where - is the smallest residue mod mmmmm. In the threefold case this criterion has a very simple form: a quotient singularity Cm/μmCm/μmC^(m)//mu_(m)\mathbb{C}^{m} / \mu_{m}Cm/μm is terminal if and only if it is of type 1m(1,−1,a)1m(1,−1,a)(1)/(m)(1,-1,a)\frac{1}{m}(1,-1, a)1m(1,−1,a), where gcd(m,a)=1gcdâ¡(m,a)=1gcd(m,a)=1\operatorname{gcd}(m, a)=1gcdâ¡(m,a)=1. This is a cyclic quotient terminal singularity.
Example ( [43,66][43,66][43,66][43,66][43,66] ). Let the cyclic group μmμmmu_(m)\boldsymbol{\mu}_{m}μm act on C4C4C^(4)\mathbb{C}^{4}C4 diagonally with weights (1,−1,a,0)(1,−1,a,0)(1,-1,a,0)(1,-1, a, 0)(1,−1,a,0), where gcd(m,a)=1gcdâ¡(m,a)=1gcd(m,a)=1\operatorname{gcd}(m, a)=1gcdâ¡(m,a)=1. Then for a polynomial ϕ(u,v)Ï•(u,v)phi(u,v)\phi(u, v)Ï•(u,v), the singularity at 0 of the quotient
is terminal whenever it is isolated. The index of this singularity equals mmmmm.
As a consequence of the classification, we obtain that the local fundamental group of the (analytic) germ of a three-dimensional terminal singularity of index mmmmm is cyclic of order mmmmm :
(1.2.1)π1(X∖{P})≃Z/mZ(1.2.1)Ï€1(X∖{P})≃Z/mZ{:(1.2.1)pi_(1)(X\\{P})≃Z//mZ:}\begin{equation*}
\pi_{1}(X \backslash\{P\}) \simeq \mathbb{Z} / m \mathbb{Z} \tag{1.2.1}
\end{equation*}(1.2.1)π1(X∖{P})≃Z/mZ
In particular, for any Weil QQQ\mathbb{Q}Q-Cartier divisor DDDDD on XXXXX, its mmmmm th multiple mDmDmDm DmD is Cartier [32, LEMMA 5.1].
The class of canonical threefold singularities is much larger than the class of terminal ones. However, there are certain boundedness results. For example, it is known that the index of a strictly canonical isolated singularity is at most 6 [31].
The modern higher-dimensional MMP often works with pairs, and one needs to extend Definition 1.1 to a wider class of objects.
Definition. Let XXXXX be a normal variety and let BBBBB be an effective QQQ\mathbb{Q}Q-divisor on XXXXX. The pair (X,B)(X,B)(X,B)(X, B)(X,B) is said to be pltpltpltp l tplt (resp. lc) if KX+BKX+BK_(X)+BK_{X}+BKX+B is QQQ\mathbb{Q}Q-Cartier and, for any birational morphism f:Y→Xf:Y→Xf:Y rarr Xf: Y \rightarrow Xf:Y→X, one can write
where BYBYB_(Y)B_{Y}BY is the proper transform of B,EiB,EiB,E_(i)B, E_{i}B,Ei are all the exceptional divisors and ai>−1ai>−1a_(i) > -1a_{i}>-1ai>−1 (resp. ai≥−1)ai≥−1{:a_(i) >= -1)\left.a_{i} \geq-1\right)ai≥−1) for all iiiii. The pair (X,B)(X,B)(X,B)(X, B)(X,B) is said to be kltkltkltk l tklt if it is plt and ⌊B⌋=0⌊B⌋=0|__ B __|=0\lfloor B\rfloor=0⌊B⌋=0.
2. MINIMAL MODEL PROGRAM
Basic elementary operations in the MMP are Mori contractions.
A contraction is a proper surjective morphism f:X→Zf:X→Zf:X rarr Zf: X \rightarrow Zf:X→Z of normal varieties with connected fibers. The exceptional locus of a contraction fffff is the subset Exc(f)⊂XExcâ¡(f)⊂XExc(f)sub X\operatorname{Exc}(f) \subset XExcâ¡(f)⊂X of points at which fffff is not an isomorphism. A Mori contraction is a contraction f:X→Zf:X→Zf:X rarr Zf: X \rightarrow Zf:X→Z such that the variety XXXXX has at worst terminal QQQ\mathbb{Q}Q-factorial singularities, the anticanonical class −KX−KX-K_(X)-K_{X}−KX is fffff-ample, and the relative Picard number ρ(X/Z)Ï(X/Z)rho(X//Z)\rho(X / Z)Ï(X/Z) equals 1. A Mori contraction is said to be divisorial (resp. flipping) if it is birational and the locus Exc(f)Excâ¡(f)Exc(f)\operatorname{Exc}(f)Excâ¡(f) has codimension 1 (resp. ≥2≥2>= 2\geq 2≥2 ). For a divisorial contraction, the exceptional locus Exc(f)Excâ¡(f)Exc(f)\operatorname{Exc}(f)Excâ¡(f) is a prime divisor. A Mori contraction whose target is a lower-dimensional variety is called a Mori fiber space. Then the general fiber is a Fano variety with at worst terminal singularities. In the particular cases where the relative dimension of X/ZX/ZX//ZX / ZX/Z equals 1 (resp. 2), the Mori fiber space f:X→Zf:X→Zf:X rarr Zf: X \rightarrow Zf:X→Z is called a QQQ\mathbb{Q}Q-conic bundle (resp. QQQ\mathbb{Q}Q-del Pezzo fibration). If ZZZZZ is a point, then XXXXX is a Fano variety with at worst terminal QQQ\mathbb{Q}Q-factorial singularities and Pic(X)≃ZPicâ¡(X)≃ZPic(X)≃Z\operatorname{Pic}(X) \simeq \mathbb{Z}Picâ¡(X)≃Z. For short, we call such varieties QQQ\mathbb{Q}Q-Fano.
The MMP procedure is a sequence of elementary transformations which are constructed inductively [35,39][35,39][35,39][35,39][35,39]. Let XXXXX be a projective algebraic variety with terminal QQQ\mathbb{Q}Q-factorial singularities. If the canonical divisor KXKXK_(X)K_{X}KX is not nef, then there exists a Mori contraction f:X→Zf:X→Zf:X rarr Zf: X \rightarrow Zf:X→Z. If fffff is divisorial, then ZZZZZ is again a variety with terminal QQQ\mathbb{Q}Q-factorial singularities and, in this situation, we can proceed with the MMP replacing XXXXX with ZZZZZ. In contrast,
a flipping contraction takes us out the category of terminal QQQ\mathbb{Q}Q-factorial varieties. To proceed, one has to perform a surgery operation as follows:
where f+f+f^(+)f^{+}f+is a contraction whose exceptional locus has codimension ≥2≥2>= 2\geq 2≥2 and the divisor KX+KX+K_(X^(+))K_{X^{+}}KX+is QQQ\mathbb{Q}Q-Cartier and f+f+f^(+)f^{+}f+-ample. Then the variety X+X+X^(+)X^{+}X+again has terminal QQQ\mathbb{Q}Q-factorial singularities, and we can proceed by replacing XXXXX with X+X+X^(+)X^{+}X+.
The process described above should terminate, and at the end we obtain a variety X¯X¯bar(X)\bar{X}X¯ such that either X¯X¯bar(X)\bar{X}X¯ has a Mori fiber space structure X¯→Z¯X¯→Z¯bar(X)rarr bar(Z)\bar{X} \rightarrow \bar{Z}X¯→Z¯ or KX¯KX¯K_( bar(X))K_{\bar{X}}KX¯ is nef. One of the remaining open problems is the termination of the program, to be more precise, termination of a sequence of flips. The problem was solved affirmatively in dimension ≤4≤4<= 4\leq 4≤4 [35,69], for varieties of general type, for uniruled varieties [5], and in some other special cases. We refer to [3] for more comprehensive survey of the higher-dimensional MMP.
The MMP has a huge number of applications in algebraic geometry. The most impressive consequence of the MMP is the finite generation of the canonical ring
of a smooth projective variety X[5,15]X[5,15]X[5,15]X[5,15]X[5,15]. Another application of the MMP is the so-called Sarkisov program which allows decomposing birational maps between Mori fiber spaces into composition of elementary transformations, called Sarkisov links [9,16,68]. Also the MMP can be applied to varieties with finite group actions and to varieties over nonclosed fields (see [63]).
As was explained above, the Mori contractions are fundamental building blocks in the MMP. To apply the MMP effectively, one needs to understand the structure of its steps in details. For a Mori contraction f:X→Zf:X→Zf:X rarr Zf: X \rightarrow Zf:X→Z of a three-dimensional variety XXXXX, there are only the following possibilities:
fffff is divisorial and the image of the (prime) exceptional divisor E:=Exc(f)E:=Excâ¡(f)E:=Exc(f)E:=\operatorname{Exc}(f)E:=Excâ¡(f) is either a point or an irreducible curve,
fffff is flipping and the exceptional locus Exc(f)Excâ¡(f)Exc(f)\operatorname{Exc}(f)Excâ¡(f) is a union of a finite number of irreducible curves,
ZZZZZ is a surface and fffff is a QQQ\mathbb{Q}Q-conic bundle,
ZZZZZ is a curve and fffff is a QQQ\mathbb{Q}Q-del Pezzo fibration,
ZZZZZ is a point and XXXXX is a QQQ\mathbb{Q}Q-Fano threefold.
Mori contractions of smooth threefolds to varieties of positive dimension where classified in the pioneering work of S. Mori [42]. S. Cutkosky [12] extended this classification to the case of Gorenstein terminal varieties. Smooth Fano threefolds of Picard number 1 where classified by Iskovskikh [22, 23] (see also [25]). Fano threefolds with Gorenstein terminal singularities are degenerated smooth ones [57]. Below we are going to discuss Mori contractions
of threefolds. We are interested only in the biregular structure of a contraction f:X→Zf:X→Zf:X rarr Zf: X \rightarrow Zf:X→Z near a fixed fiber f−1(o),o∈Zf−1(o),o∈Zf^(-1)(o),o in Zf^{-1}(o), o \in Zf−1(o),o∈Z. Typically, we do not consider the simple case where XXXXX is Gorenstein.
3. GENERAL ELEPHANT
A natural way to study higher-dimensional varieties is the inductive one. Typically, to apply this method, we need to find a certain subvariety of dimension one less (divisor) which is sufficiently good is the sense of singularities.
Conjecture 3.1. Let f:X→(Z∋o)f:X→(Z∋o)f:X rarr(Z∋o)f: X \rightarrow(Z \ni o)f:X→(Z∋o) be a threefold Mori contraction, where (Z∋o)(Z∋o)(Z∋o)(Z \ni o)(Z∋o) is a small neighborhood. Then the general member D∈|−KX|D∈−KXD in|-K_(X)|D \in\left|-K_{X}\right|D∈|−KX| is a normal surface with Du Val singularities.
The conjecture was proposed by M. Reid who called a good member of |−KX|−KX|-K_(X)|\left|-K_{X}\right||−KX| "elephant." We follow this language and call Conjecture 3.1 the General Elephant Conjecture. The importance of the existence of a good member in |−KX|−KX|-K_(X)|\left|-K_{X}\right||−KX| is motivated by many reasons:
The general elephant passes through all the non-Gorenstein points of XXXXX and so it encodes the information about their types and configuration (cf. Proposition 3.2 below).
For flipping contractions, Conjecture 3.1 is a sufficient condition for the existence of threefold flips [32].
For a divisorial contraction f:X→Zf:X→Zf:X rarr Zf: X \rightarrow Zf:X→Z whose fibers have dimension ≤1≤1<= 1\leq 1≤1, the image DZ:=f(D)DZ:=f(D)D_(Z):=f(D)D_{Z}:=f(D)DZ:=f(D) of a Du Val elephant D∈|−KX|D∈−KXD in|-K_(X)|D \in\left|-K_{X}\right|D∈|−KX| must be again Du Val and the image Γ:=f(E)Γ:=f(E)Gamma:=f(E)\Gamma:=f(E)Γ:=f(E) of the exceptional divisor is a curve on DZDZD_(Z)D_{Z}DZ. Then one can reconstruct fffff starting from the triple (Z⊃DZ⊃Γ)Z⊃DZ⊃Γ(Z supD_(Z)sup Gamma)\left(Z \supset D_{Z} \supset \Gamma\right)(Z⊃DZ⊃Γ) by using a certain birational procedure. Such an approach was successfully worked out in many cases by N. Tziolas [71-74].
If f:X→(Z∋o)f:X→(Z∋o)f:X rarr(Z∋o)f: X \rightarrow(Z \ni o)f:X→(Z∋o) is a QQQ\mathbb{Q}Q-del Pezzo fibration such that general D∈|−KX|D∈−KXD in|-K_(X)|D \in\left|-K_{X}\right|D∈|−KX| is Du Val, then, composing the projection D→ZD→ZD rarr ZD \rightarrow ZD→Z with minimal resolution D~→DD~→Dtilde(D)rarr D\tilde{D} \rightarrow DD~→D, we obtain a relatively minimal elliptic fibration whose singular fibers are classified by Kodaira [36]. Then one can get a bound of multiplicities of fibers and describe the configuration of non-Gorenstein singularities.
For a QQQ\mathbb{Q}Q-Fano threefold XXXXX, a Du Val general elephant is a (singular) K3 surface. In the case where the linear system |−KX|−KX|-K_(X)|\left|-K_{X}\right||−KX| is "sufficiently big," this implies the existence of a good Gorenstein model [1].
Shokurov [70] generalized Conjecture 3.1 and introduced a new notion which is very efficient in the study of pluri-anticanonical linear systems. Omitting technicalities, we reproduce a weak form of Shokurov's definition.
Definition. An nnnnn-complement of the canonical class KXKXK_(X)K_{X}KX is a member D∈|−nKX|D∈−nKXD in|-nK_(X)|D \in\left|-n K_{X}\right|D∈|−nKX| such that the pair (X,1nD)X,1nD(X,(1)/(n)D)\left(X, \frac{1}{n} D\right)(X,1nD) is lc. An nnnnn-complement is said to be klt (resp. plt) if such is the pair (X,1nD)X,1nD(X,(1)/(n)D)\left(X, \frac{1}{n} D\right)(X,1nD).
According to the inversion of adjunction [70], the existence of a Du Val general elephant D∈|−KX|D∈−KXD in|-K_(X)|D \in\left|-K_{X}\right|D∈|−KX| is equivalent to the existence of a plt 1-complement. Shokurov developed a powerful theory that works in arbitrary dimension and allows constructing complements inductively (see [64,70][64,70][64,70][64,70][64,70] and references therein).
Note that Reid's general elephant fails for Fano threefolds. For example, in [6,21][6,21][6,21][6,21][6,21] one can find examples of QQQ\mathbb{Q}Q-Fano threefolds with an empty anticanonical linear system. Because of this, the statement of Conjecture 3.1 sometimes is called a "principle." Nonetheless, there are only a few examples of such Fano threefolds. In the case dim(Z)>0dimâ¡(Z)>0dim(Z) > 0\operatorname{dim}(Z)>0dimâ¡(Z)>0, Conjecture 3.1 is expected to be true without exceptions. The following should be considered as the local version of Conjecture 3.1.
Proposition 3.2 (Reid [66]). Let (X∋P(X∋P(X∋P(X \ni P(X∋P ) be the analytic germ of a threefold terminal singularity of index m>1m>1m > 1m>1m>1. Then the general member D∈|−KX|D∈−KXD in|-K_(X)|D \in\left|-K_{X}\right|D∈|−KX| is a Du Val singularity. Furthermore, let π:X′→XÏ€:X′→Xpi:X^(')rarr X\pi: X^{\prime} \rightarrow XÏ€:X′→X be the index-one cover and let D′:=π−1(D)D′:=π−1(D)D^('):=pi^(-1)(D)D^{\prime}:=\pi^{-1}(D)D′:=π−1(D). Then the cover D′→DD′→DD^(')rarr DD^{\prime} \rightarrow DD′→D belongs to one of the following six types:
In this section we treat divisorial Mori contractions of a divisor to a point. Such contractions are studied very well due to works of Y. Kawamata [34], A. Corti [10], M. Kawakita [26-30], T. Hayakawa [18-20], and others. In this case, General Elephant Conjecture 3.1 has been verified:
Theorem 4.1 (Kawakita [28,29]). Let f:X→(Z∋o)f:X→(Z∋o)f:X rarr(Z∋o)f: X \rightarrow(Z \ni o)f:X→(Z∋o) be a divisorial Mori contraction that contracts a divisor to a point. Then the general member D∈|−KX|D∈−KXD in|-K_(X)|D \in\left|-K_{X}\right|D∈|−KX| is Du Val.
One of the main tools in the proofs is the orbifold Riemann-Roch formula [66]: if XXXXX is a three-dimensional projective variety with terminal singularities and DDDDD is a Weil QQQ\mathbb{Q}Q-Cartier divisor on XXXXX, then for the sheaf L=OX(D)L=OX(D)L=O_(X)(D)\mathscr{L}=\mathscr{O}_{X}(D)L=OX(D) there is a formula of the form
(4.1.1)χ(L)=χ(OX)+112D⋅(D−KX)⋅(2D−KX)+112D⋅c2+∑PcP(D)(4.1.1)χ(L)=χOX+112Dâ‹…D−KXâ‹…2D−KX+112Dâ‹…c2+∑P cP(D){:(4.1.1)chi(L)=chi(O_(X))+(1)/(12)D*(D-K_(X))*(2D-K_(X))+(1)/(12)D*c_(2)+sum_(P)c_(P)(D):}\begin{equation*}
\chi(\mathscr{L})=\chi\left(\mathscr{O}_{X}\right)+\frac{1}{12} D \cdot\left(D-K_{X}\right) \cdot\left(2 D-K_{X}\right)+\frac{1}{12} D \cdot \mathrm{c}_{2}+\sum_{P} c_{P}(D) \tag{4.1.1}
\end{equation*}(4.1.1)χ(L)=χ(OX)+112D⋅(D−KX)⋅(2D−KX)+112D⋅c2+∑PcP(D)
where the sum rungs over all the virtual quotient singularities of XXXXX, i.e., over the actual singularities of XXXXX that are replaced with their small deformations [66], and cP(D)cP(D)c_(P)(D)c_{P}(D)cP(D) is a local
contribution due to singularity at PPPPP, depending only on the local analytic type of DDDDD at PPPPP. There is an explicit formula for the computation of cP(D)cP(D)c_(P)(D)c_{P}(D)cP(D).
Except for a few hard cases, the classification of divisorial Mori contractions of a divisor to a point has been completed. A typical result here is to show that a contraction is a weighted blowup with some explicit collection of weights:
Theorem 4.2 (Y. Kawamata [34]). Let f:X→(Z∋o)f:X→(Z∋o)f:X rarr(Z∋o)f: X \rightarrow(Z \ni o)f:X→(Z∋o) be a divisorial Mori contraction that contracts a divisor to a point. Assume that o∈Zo∈Zo in Zo \in Zo∈Z is a cyclic quotient singularity of type 1r(a,−a,1)1r(a,−a,1)(1)/(r)(a,-a,1)\frac{1}{r}(a,-a, 1)1r(a,−a,1). Then fffff is the weighted blowup with weights (a/r,1−a/r,1/r)(a/r,1−a/r,1/r)(a//r,1-a//r,1//r)(a / r, 1-a / r, 1 / r)(a/r,1−a/r,1/r).
Theorem 4.3 (M. Kawakita [26]). Let f:X→(Z∋o)f:X→(Z∋o)f:X rarr(Z∋o)f: X \rightarrow(Z \ni o)f:X→(Z∋o) be a divisorial Mori contraction that contracts a divisor to a smooth point. Then fffff is the weighted blowup with weights (1,a,b)(1,a,b)(1,a,b)(1, a, b)(1,a,b), where gcd(a,b)=1gcdâ¡(a,b)=1gcd(a,b)=1\operatorname{gcd}(a, b)=1gcdâ¡(a,b)=1.
These results are intensively used in the three-dimensional birational geometry, for example, in the proof of birational rigidity of index-1 Fano threefold weighted hypersurfaces [11][11][11][11][11].
5. DEL PEZZO FIBRATIONS
Much less is known about the local structure of QQQ\mathbb{Q}Q-del Pezzo fibrations. As was explained in Section 3, the existence of a Du Val general elephant would be very helpful in the study such contractions. However, in this case Conjecture 3.1 is established only in some special situations.
An important question that can be asked in the Del Pezzo fibration case is the presence of multiple fibers.
Theorem 5.1 ([49]). Let f:X→Zf:X→Zf:X rarr Zf: X \rightarrow Zf:X→Z be a QQQ\mathbb{Q}Q-del Pezzo fibration and let f∗(o)=moFof∗(o)=moFof^(**)(o)=m_(o)F_(o)f^{*}(o)=m_{o} F_{o}f∗(o)=moFo be a special fiber of multiplicity momom_(o)m_{o}mo. Then mo≤6mo≤6m_(o) <= 6m_{o} \leq 6mo≤6 and all the cases 1≤mo≤61≤mo≤61 <= m_(o) <= 61 \leq m_{o} \leq 61≤mo≤6 occur. Moreover, the possibilities for the local behavior of FoFoF_(o)F_{o}Fo near singular points are described.
The main idea of the proof is to apply the orbifold Riemann-Roch formula (4.1.1) to the divisor FoFoF_(o)F_{o}Fo and its multiples.
Example. Suppose that the cyclic group μ4μ4mu_(4)\mu_{4}μ4 acts on Px1×Py1×CtPx1×Py1×CtP_(x)^(1)xxP_(y)^(1)xxC_(t)\mathbb{P}_{x}^{1} \times \mathbb{P}_{y}^{1} \times \mathbb{C}_{t}Px1×Py1×Ct via
(x,y;t)⟼(y,−x,−1t)(x,y;t)⟼(y,−x,−1t)(x,y;t)longmapsto(y,-x,sqrt(-1)t)(x, y ; t) \longmapsto(y,-x, \sqrt{-1} t)(x,y;t)⟼(y,−x,−1t)
is the germ of a QQQ\mathbb{Q}Q-del Pezzo fibration with central fiber of multiplicity 4.
Another type of QQQ\mathbb{Q}Q-del Pezzo fibrations which are investigated relatively well are those whose central fiber F:=f−1(o)F:=f−1(o)F:=f^(-1)(o)F:=f^{-1}(o)F:=f−1(o) is reduced, normal, and has "good" singularities. Then XXXXX can be viewed as a one-parameter smoothing of FFFFF. The total space of this smoothing must be QQQ\mathbb{Q}Q-Gorenstein and FFFFF can be viewed as a degeneration of a general fiber (smooth del Pezzo surface) in a QQQ\mathbb{Q}Q-Gorenstein family. The most studied class of singularities admitting QQQ\mathbb{Q}Q-Gorenstein smoothings is the class of singularities of type T.
Definition (Kollár, Shepherd-Barron [40]). A two-dimensional quotient singularity is said to be of type TTT\mathrm{T}T if it admits a smoothing in a one-parameter QQQ\mathbb{Q}Q-Gorenstein family X→BX→BX rarr BX \rightarrow BX→B.
In this case, by the inversion of adjunction [70], the pair (X,F)(X,F)(X,F)(X, F)(X,F) is plt and the total family XXXXX is terminal. Conversely, if X∋PX∋PX∋PX \ni PX∋P is a QQQ\mathbb{Q}Q-Gorenstein point and FFFFF is an effective Cartier divisor at PPPPP such that the pair (X,F)(X,F)(X,F)(X, F)(X,F) is plt, then F∋PF∋PF∋PF \ni PF∋P is a T-singularity and the point X∋PX∋PX∋PX \ni PX∋P is terminal. Singularities of type T and their deformations were studied by Kollár and Shepherd-Barron [40]. In particular, they proved that any T-singularity is either a Du Val point or a cyclic quotient of type 1m(q1,q2)1mq1,q2(1)/(m)(q_(1),q_(2))\frac{1}{m}\left(q_{1}, q_{2}\right)1m(q1,q2) with
Minimal resolutions of these singularities are also described [40, § 3].
Thus to study QQQ\mathbb{Q}Q-del Pezzo fibrations whose central fiber has only quotient singularities, one has to consider QQQ\mathbb{Q}Q-Gorenstein smoothings of del Pezzo surfaces with singularities of type T. The important auxiliary fact here is the unobstructedness of deformations:
Proposition 5.2 ( [13,41])[13,41])[13,41])[13,41])[13,41]). Let FFFFF be a projective surface with log canonical singularities such that −KF−KF-K_(F)-K_{F}−KF is big. Then there are no local-to-global obstructions to deformations of FFFFF. In particular, if FFFFF has TTT\mathrm{T}T-singularities, then FFFFF admits a QQQ\mathbb{Q}Q-Gorenstein smoothing.
Theorem 5.3 (Hacking-Prokhorov [13]). Let F be a projective surface with quotient singularities such that −KF−KF-K_(F)-K_{F}−KF is ample, ρ(F)=1Ï(F)=1rho(F)=1\rho(F)=1Ï(F)=1, and FFFFF admits a QQQ\mathbb{Q}Q-Gorenstein smoothing. Then FFFFF belongs to one of the following:
14 infinite sequences of toric surfaces (see below);
partial smoothing of a toric surface as above;
18 sporadic families of surfaces of index ≤2≤2<= 2\leq 2≤2 [2].
Toric surfaces appearing in the above theorem are determined by a Markov-type equation. More precisely, for KF2≥5KF2≥5K_(F)^(2) >= 5K_{F}^{2} \geq 5KF2≥5 these surfaces are weighted projective spaces given by the following table:
a2+b2+5c2=5abca2+b2+5c2=5abca^(2)+b^(2)+5c^(2)=5abca^{2}+b^{2}+5 c^{2}=5 a b ca2+b2+5c2=5abc
K_(F)^(2) F Markov-type equation
9 P(a^(2),b^(2),c^(2)) a^(2)+b^(2)+c^(2)=3abc
8 P(a^(2),b^(2),2c^(2)) a^(2)+b^(2)+2c^(2)=4abc
6 P(a^(2),2b^(2),3c^(2)) a^(2)+2b^(2)+3c^(2)=6abc
5 P(a^(2),b^(2),5c^(2)) a^(2)+b^(2)+5c^(2)=5abc| $K_{F}^{2}$ | $F$ | Markov-type equation |
| ---: | :--- | :--- |
| 9 | $\mathbb{P}\left(a^{2}, b^{2}, c^{2}\right)$ | $a^{2}+b^{2}+c^{2}=3 a b c$ |
| 8 | $\mathbb{P}\left(a^{2}, b^{2}, 2 c^{2}\right)$ | $a^{2}+b^{2}+2 c^{2}=4 a b c$ |
| 6 | $\mathbb{P}\left(a^{2}, 2 b^{2}, 3 c^{2}\right)$ | $a^{2}+2 b^{2}+3 c^{2}=6 a b c$ |
| 5 | $\mathbb{P}\left(a^{2}, b^{2}, 5 c^{2}\right)$ | $a^{2}+b^{2}+5 c^{2}=5 a b c$ |
and for K2≤4K2≤4K^(2) <= 4K^{2} \leq 4K2≤4 they are certain abelian quotients of the weighted projective spaces as above. Note, however, that in general we cannot assert that, for central fiber FFFFF of a QQQ\mathbb{Q}Q-del Pezzo fibration, the condition ρ(F)=1Ï(F)=1rho(F)=1\rho(F)=1Ï(F)=1 holds. Some partial results in the case ρ(F)>1Ï(F)>1rho(F) > 1\rho(F)>1Ï(F)>1 where obtained in [60]. In particular, [60] establishes the existence of Du Val general elephant for QQQ\mathbb{Q}Q-del Pezzo fibrations with "good" fibers:
Theorem 5.4. Let f:X→(Z∋o)f:X→(Z∋o)f:X rarr(Z∋o)f: X \rightarrow(Z \ni o)f:X→(Z∋o) be a QQQ\mathbb{Q}Q-del Pezzo fibration over a curve germ Z∋oZ∋oZ∋oZ \ni oZ∋o. Assume that the fiber f−1(o)f−1(o)f^(-1)(o)f^{-1}(o)f−1(o) is reduced, normal, and has only log terminal singularities. Then the general elephant D∈|−KX|D∈−KXD in|-K_(X)|D \in\left|-K_{X}\right|D∈|−KX| is Du Val.
Theorem 5.3 gives a complete answer to the question posed by M. Manetti [41]:
Corollary 5.5 ([13]). Let XXXXX be a projective surface with quotient singularities which admits a smoothing to P2P2P^(2)\mathbb{P}^{2}P2. Then XXXXX is a QQQ\mathbb{Q}Q-Gorenstein deformation of a weighted projective plane P(a2,b2,c2)Pa2,b2,c2P(a^(2),b^(2),c^(2))\mathbb{P}\left(a^{2}, b^{2}, c^{2}\right)P(a2,b2,c2), where the triple (a,b,c)(a,b,c)(a,b,c)(a, b, c)(a,b,c) is a solution of the Markov equation
a2+b2+c2=3abca2+b2+c2=3abca^(2)+b^(2)+c^(2)=3abca^{2}+b^{2}+c^{2}=3 a b ca2+b2+c2=3abc
Results similar to Theorem 5.3 were obtained for QQQ\mathbb{Q}Q-del Pezzo fibrations whose central fiber is log canonical [62]. However, in this case the classification is not complete.
6. EXTREMAL CURVE GERMS
To study Mori contractions with fibers of dimension ≤1≤1<= 1\leq 1≤1, it is convenient to work with analytic threefolds and to localize to situation near a curve contained in a fiber.
Definition 6.1. Let (X⊃C)(X⊃C)(X sup C)(X \supset C)(X⊃C) be the analytic germ of a threefold with terminal singularities along a reduced connected complete curve. Then (X⊃C)(X⊃C)(X sup C)(X \supset C)(X⊃C) is called an extremal curve germ if there exists a contraction
f:(X⊃C)⟶(Z∋o)f:(X⊃C)⟶(Z∋o)f:(X sup C)longrightarrow(Z∋o)f:(X \supset C) \longrightarrow(Z \ni o)f:(X⊃C)⟶(Z∋o)
such that C=f−1(o)red C=f−1(o)red C=f^(-1)(o)_("red ")C=f^{-1}(o)_{\text {red }}C=f−1(o)red and −KX−KX-K_(X)-K_{X}−KX is fffff-ample. The curve CCCCC is called the central fiber of the germ and Z∋oZ∋oZ∋oZ \ni oZ∋o is called the target variety or the base of (X⊃C)(X⊃C)(X sup C)(X \supset C)(X⊃C). An extremal curve germ is said to be irreducible if such is its central fiber.
In the definition above, we do not assume that XXXXX is QQQ\mathbb{Q}Q-factorial or ρ(X/Z)=1Ï(X/Z)=1rho(X//Z)=1\rho(X / Z)=1Ï(X/Z)=1. This is because QQQ\mathbb{Q}Q-factoriality typically is not a local condition in the analytic category (see [32,$1])[32,$1])[32,$1])[32, \$ 1])[32,$1]). There are three types of extremal curve germs:
flipping if fffff is birational and does not contract divisors;
divisorial if the exceptional locus is two-dimensional;
QQQ\mathbb{Q}Q-conic bundle germ if the target variety ZZZZZ is a surface.
If a divisorial curve germ is irreducible, then the exceptional locus of the corresponding contraction is a QQQ\mathbb{Q}Q-Cartier divisor and the target variety ZZZZZ has terminal singularities [51, §3]. In general, this is not true. It may happen that the exceptional locus is a union of a divisor and some curves.
As an example, we consider the case where XXXXX has singularities of indices 1 and 2 .
Theorem 6.2 ([47]). Let (X⊃C)(X⊃C)(X sup C)(X \supset C)(X⊃C) be a QQQ\mathbb{Q}Q-conic bundle germ over a smooth base. Assume that XXXXX is not Gorenstein and 2KX2KX2K_(X)2 K_{X}2KX is Cartier. Then XXXXX can be embedded to P(1,1,1,2)×C2P(1,1,1,2)×C2P(1,1,1,2)xxC^(2)\mathbb{P}(1,1,1,2) \times \mathbb{C}^{2}P(1,1,1,2)×C2
and given there by two quadratic equations. In particular, the point P∈XP∈XP in XP \in XP∈X of index 2 is unique, the curve CCCCC has at most 4 components, all of them pass through PPPPP.
Theorem 6.3 ([38]). Let (X⊃C)(X⊃C)(X sup C)(X \supset C)(X⊃C) be a flipping extremal curve germ and let
be the corresponding flip. Assume that 2KX2KX2K_(X)2 K_{X}2KX is Cartier. Then (Z∋o)(Z∋o)(Z∋o)(Z \ni o)(Z∋o) is the quotient of the isolated hypersurface singularity
by the μ2μ2mu_(2)\boldsymbol{\mu}_{2}μ2-action given by the weights (1,1,0,0)(1,1,0,0)(1,1,0,0)(1,1,0,0)(1,1,0,0). The contraction fffff (resp. f+f+f^(+)f^{+}f+) is the quotient of the blowup of the plane {x2=x3=0}x2=x3=0{x_(2)=x_(3)=0}\left\{x_{2}=x_{3}=0\right\}{x2=x3=0} (resp. {x1=x2=0}x1=x2=0{x_(1)=x_(2)=0}\left\{x_{1}=x_{2}=0\right\}{x1=x2=0} ) by μ2μ2mu_(2)\mu_{2}μ2. In particular, XXXXX contains a unique point of index 2 and the central fiber CCCCC is irreducible. The variety X+X+X^(+)X^{+}X+ is Gorenstein.
A similar description is known for divisorial extremal curve germs of index 2 [38,84][38,84][38,84][38,84][38,84].
First properties. Let (X⊃C)(X⊃C)(X sup C)(X \supset C)(X⊃C) be an extremal curve germ and let f:(X⊃C)→(Z∋o)f:(X⊃C)→(Z∋o)f:(X sup C)rarr(Z∋o)f:(X \supset C) \rightarrow(Z \ni o)f:(X⊃C)→(Z∋o) be the corresponding contraction. For any connected subcurve C′⊂CC′⊂CC^(')sub CC^{\prime} \subset CC′⊂C, the germ (X⊃C′)X⊃C′(X supC^('))\left(X \supset C^{\prime}\right)(X⊃C′) is again an extremal curve germ. If, moreover, C′⫋CC′⫋CC^(')â«‹CC^{\prime} \varsubsetneqq CC′⫋C, then (X⊃C′)X⊃C′(X supC^('))\left(X \supset C^{\prime}\right)(X⊃C′) is birational. By the Kawamata-Viehweg vanishing theorem,
(see, e.g., [35]). As a consequence, one has pa(C′)≤0paC′≤0p_(a)(C^(')) <= 0\mathrm{p}_{\mathrm{a}}\left(C^{\prime}\right) \leq 0pa(C′)≤0 for any subcurve C′⊂CC′⊂CC^(')sub CC^{\prime} \subset CC′⊂C. In particular, C=⋃CiC=⋃CiC=uuuC_(i)C=\bigcup C_{i}C=⋃Ci is a "tree" of smooth rational curves. Furthermore,
where nnnnn is the number of irreducible components of CCCCC. For more delicate properties of extremal curve germs, one needs to know the cohomology of the dualizing sheaf, see [44,47]:
(6.3.3)R1f∗ωX={0, if f is birational, ωZ, if f is Q-conic bundle and Z is smooth. (6.3.3)R1f∗ωX=0, if f is birational, ωZ, if f is Q-conic bundle and Z is smooth. {:(6.3.3)R^(1)f_(**)omega_(X)={[0","," if "f" is birational, "],[omega_(Z)","," if "f" is "Q"-conic bundle and "Z" is smooth. "]:}:}R^{1} f_{*} \omega_{X}= \begin{cases}0, & \text { if } f \text { is birational, } \tag{6.3.3}\\ \omega_{Z}, & \text { if } f \text { is } \mathbb{Q} \text {-conic bundle and } Z \text { is smooth. }\end{cases}(6.3.3)R1f∗ωX={0, if f is birational, ωZ, if f is Q-conic bundle and Z is smooth.Â
Definition. An irreducible extremal curve germ (X⊃C)(X⊃C)(X sup C)(X \supset C)(X⊃C) is (locally) imprimitive at a point PPPPP if the inverse image of CCCCC under the index-one cover (X♯∋P♯)→(X∋P)X♯∋P♯→(X∋P)(X^(♯)∋P^(♯))rarr(X∋P)\left(X^{\sharp} \ni P^{\sharp}\right) \rightarrow(X \ni P)(X♯∋P♯)→(X∋P) splits.
Theorem 6.4 ([44,47]). Let (X⊃C)(X⊃C)(X sup C)(X \supset C)(X⊃C) be an extremal curve germ and let C1,…,CnC1,…,CnC_(1),dots,C_(n)C_{1}, \ldots, C_{n}C1,…,Cn be irreducible components of CCCCC.
Each CiCiC_(i)C_{i}Ci contains at most 3 singular points of XXXXX.
Each CiCiC_(i)C_{i}Ci contains at most 2 non-Gorenstein points of XXXXX and at most 1 point which is imprimitive for (X⊃Ci)X⊃Ci(X supC_(i))\left(X \supset C_{i}\right)(X⊃Ci).
To prove the first assertion, one needs to analyze the conormal sheaf IC/IC2IC/IC2I_(C)//I_(C)^(2)I_{C} / I_{C}^{2}IC/IC2 and use the vanishing H1(OX/J)=0H1OX/J=0H^(1)(O_(X)//J)=0H^{1}\left(\mathscr{O}_{X} / J\right)=0H1(OX/J)=0 for any J⊂OXJ⊂OXJ subO_(X)J \subset \mathscr{O}_{X}J⊂OX with Supp(OX/J)=CSuppâ¡OX/J=CSupp(O_(X)//J)=C\operatorname{Supp}\left(\mathscr{O}_{X} / J\right)=CSuppâ¡(OX/J)=C (see [44,55]). For the second assertion, one can use topological arguments based on (1.2.1) (see [55]). For the last assertion, we refer to [44, 1.15], [37, 4.2], and [55, 4.7.6]
The techniques applied in the proof of the above proposition allow obtaining much stronger results. In particular, they alow classifying all the possibilities for the local behavior of an irreducible germ (X⊃C)(X⊃C)(X sup C)(X \supset C)(X⊃C) near a singular point PPPPP [44]. Thus, according to [44] and [47], the triple (X⊃C∋P(X⊃C∋P(X sup C∋P(X \supset C \ni P(X⊃C∋P ) belongs to one of the following types:
Here the symbol ∨∨^(vv){ }^{\vee}∨ means that (X⊃C∋P)(X⊃C∋P)(X sup C∋P)(X \supset C \ni P)(X⊃C∋P) is locally imprimitive, the symbol II means that ( X∋PX∋PX∋PX \ni PX∋P ) is a terminal point of exceptional type cAx/4 (see Proposition 3.2), and III means that (X∋P(X∋P(X∋P(X \ni P(X∋P ) is an (isolated) cDV-point.
For example, a triple ( X⊃C∋PX⊃C∋PX sup C∋PX \supset C \ni PX⊃C∋P ) is of type (IC) if there are analytic isomorphisms
where mmmmm is odd and m≥5m≥5m >= 5m \geq 5m≥5. For definitions other types, we refer to [44] and [47].
6.1. Construction of germs by deformations
Let (X⊃C)(X⊃C)(X sup C)(X \supset C)(X⊃C) be an extremal curve germ and let f:X→(Z∋o)f:X→(Z∋o)f:X rarr(Z∋o)f: X \rightarrow(Z \ni o)f:X→(Z∋o) be the corresponding contraction. Denote by |OZ|OZ|O_(Z)|\left|\mathscr{O}_{Z}\right||OZ| the infinite-dimensional linear system of hyperplane sections passing through ooooo and let |OX|:=f∗|OZ|OX:=f∗OZ|O_(X)|:=f^(**)|O_(Z)|\left|\mathscr{O}_{X}\right|:=f^{*}\left|\mathscr{O}_{Z}\right||OX|:=f∗|OZ|. The general hyperplane section of (X⊃CX⊃C(X sup C:}\left(X \supset C\right.(X⊃C ) is the general member H∈|OX|H∈OXH in|O_(X)|H \in\left|\mathscr{O}_{X}\right|H∈|OX|. The divisor HHHHH contains much more information on the total space than a general elephant D∈|−KX|D∈−KXD in|-K_(X)|D \in\left|-K_{X}\right|D∈|−KX|. However, the singularities of HHHHH typically are more complicated, in particular, HHHHH can be nonnormal.
The variety XXXXX (resp. ZZZZZ ) can be viewed as the total space of a one-parameter deformation of HHHHH (resp. HZ:=f(H)HZ:=f(H)H_(Z):=f(H)H_{Z}:=f(H)HZ:=f(H) ). We are going to reverse this consideration.
Construction (see [38, § 11], [44, § 1B]). Suppose we are given a normal surface germ (H⊃CH⊃C(H sup C:}\left(H \supset C\right.(H⊃C ) along a proper curve CCCCC and a contraction fH:H→HZfH:H→HZf_(H):H rarrH_(Z)f_{H}: H \rightarrow H_{Z}fH:H→HZ such that CCCCC is a fiber and −KH−KH-K_(H)-K_{H}−KH is fHfHf_(H)f_{H}fH-ample. Let P1,…,Pr∈HP1,…,Pr∈HP_(1),dots,P_(r)in HP_{1}, \ldots, P_{r} \in HP1,…,Pr∈H be all the singular points. Assume also that near each PiPiP_(i)P_{i}Pi there exists a small one-parameter deformation SiSiS_(i)\mathfrak{S}_{i}Si of a neighborhood HiHiH_(i)H_{i}Hi of PiPiP_(i)P_{i}Pi in HHHHH such that the total space SiSiS_(i)\mathfrak{S}_{i}Si has a terminal singularity at PiPiP_(i)P_{i}Pi. The obstruction to globalize deformations
Then we have a threefold X:=S⊃CX:=S⊃CX:=Ssup CX:=\mathfrak{S} \supset CX:=S⊃C with H∈|OX|H∈OXH in|O_(X)|H \in\left|\mathscr{O}_{X}\right|H∈|OX| such that locally near PiPiP_(i)P_{i}Pi it has the desired structure and one can extend fHfHf_(H)f_{H}fH to a contraction f:X→Zf:X→Zf:X rarr Zf: X \rightarrow Zf:X→Z which is birational (resp. a QQQ\mathbb{Q}Q-conic bundle) if HZHZH_(Z)H_{Z}HZ is a surface (resp. a curve).
Example. Consider a rational curve fibration fH~:H~→HZfH~:H~→HZf_( tilde(H)): tilde(H)rarrH_(Z)f_{\tilde{H}}: \tilde{H} \rightarrow H_{Z}fH~:H~→HZ over a smooth curve germ HZ∋oHZ∋oH_(Z)∋oH_{Z} \ni oHZ∋o, where H~H~tilde(H)\tilde{H}H~ is a smooth surface such that the fiber over ooooo has the following weighted dual graph:
Contracting the curves corresponding to the white vertices ◻◻◻\squareâ—» and ∘∘@\circ∘, we obtain a singular surface HHHHH and a KHKHK_(H)K_{H}KH-negative contraction fH:H→HZfH:H→HZf_(H):H rarrH_(Z)f_{H}: H \rightarrow H_{Z}fH:H→HZ whose fiber over ooooo is a curve C⊂HC⊂HC sub HC \subset HC⊂H having three irreducible components that correspond to the black vertices ∙∙∙\bullet∙. The singular locus of HHHHH consists of a Du Val point P0∈HP0∈HP_(0)in HP_{0} \in HP0∈H of type A1A1A_(1)\mathrm{A}_{1}A1 and a log canonical singularity P∈HP∈HP in HP \in HP∈H whose dual graph is formed by the white circle vertices ∘∘@\circ∘. Both P0P0P_(0)P_{0}P0 and PPPPP have 1-parameter QQQ\mathbb{Q}Q-Gorenstein smoothings [38, computation 6.7.1]. Applying the above construction to H⊃CH⊃CH sup CH \supset CH⊃C, we obtain an example of a QQQ\mathbb{Q}Q-conic bundle contraction f:(X⊃C)→(Z∋o)f:(X⊃C)→(Z∋o)f:(X sup C)rarr(Z∋o)f:(X \supset C) \rightarrow(Z \ni o)f:(X⊃C)→(Z∋o) with a unique non-Gorenstein point which is of type cD/3. If we remove the (-2)-curve corresponding to ◻◻◻\squareâ—» on the left-hand side of the graph, we get a birational contraction of surfaces fH′:H′→HZ′fH′:H′→HZ′f_(H)^('):H^(')rarrH_(Z)^(')f_{H}^{\prime}: H^{\prime} \rightarrow H_{Z}^{\prime}fH′:H′→HZ′. Applying the same construction to H′⊃CH′⊃CH^(')sup CH^{\prime} \supset CH′⊃C, we obtain an example of a divisorial contraction. Similarly, removing further one of the (−1)(−1)(-1)(-1)(−1)-curves, we get a flip.
7. EXTREMAL CURVE GERMS: GENERAL ELEPHANT
Theorem 7.1 (Mori [44], Kollár-Mori [38], Mori-Prokhorov [50]). Let (X⊃C)(X⊃C)(X sup C)(X \supset C)(X⊃C) be an irreducible extremal curve germ. Then the general member D∈|−KX|D∈−KXD in|-K_(X)|D \in\left|-K_{X}\right|D∈|−KX| has only Du Val singularities.
The existence of a Du Val elephant for extremal curve germs with reducible central fiber is not known at the moment. See Theorem 9.2 below for partial results in this direction.
Comment on the proof. Essentially, there are three methods to find a good elephant D∈|−KX|D∈−KXD in|-K_(X)|D \in\left|-K_{X}\right|D∈|−KX|. We outline them below.
Finally, in the most complicated cases, none of the above methods work. Then one needs more subtle techniques which require detailed analysis of singularities and infinitesimal structure of XXXXX along CCCCC [44, §§ 8-9]. Then, roughly speaking, the good section D∈|−KX|D∈−KXD in|-K_(X)|D \in\left|-K_{X}\right|D∈|−KX| is recovered as the formal Weil divisor limCnlimCnlimC_(n)\lim C_{n}limCn of the completion X∧X∧X^X^{\wedge}X∧ of XXXXX along CCCCC, where CnCnC_(n)C_{n}Cn are subschemes with support CCCCC constructed by using certain inductive procedure [44,§9][44,§9][44,§9][44, \S 9]§[44,§9].
As a consequence of Theorem 7.1, in the QQQ\mathbb{Q}Q-conic bundle case, one obtains the following fact which proves Iskovskikh's conjecture [24].
Corollary 7.2. Let (X⊃C)(X⊃C)(X sup C)(X \supset C)(X⊃C) be a QQQ\mathbb{Q}Q-conic bundle germ over (Z∋o)(Z∋o)(Z∋o)(Z \ni o)(Z∋o), where CCCCC can be reducible. Then (Z∋o)(Z∋o)(Z∋o)(Z \ni o)(Z∋o) is a Du Val singularity of type AnAnA_(n)\mathrm{A}_{\mathrm{n}}An (or smooth).
This result is very useful for applications to the rationality problem of three-dimensional varieties having conic bundle structure [24,61][24,61][24,61][24,61][24,61] and some problems of biregular geometry [58,59][58,59][58,59][58,59][58,59].
It turns out that the structure of QQQ\mathbb{Q}Q-conic bundle germs over a singular base ( Z∋oZ∋oZ∋oZ \ni oZ∋o ) is much simpler and shorter than others. In fact, these germs can be exhibited as certain quotients of QQQ\mathbb{Q}Q-conic bundles of index ≤2≤2<= 2\leq 2≤2 (see Theorem 6.2). A complete classification of such germs was obtained in [47,48][47,48][47,48][47,48][47,48]. Here is a typical example.
Example 7.3. Let the group μnμnmu_(n)\mu_{n}μn act on Cu,v2Cu,v2C_(u,v)^(2)\mathbb{C}_{u, v}^{2}Cu,v2 and Px,y1×Cu,v2Px,y1×Cu,v2P_(x,y)^(1)xxC_(u,v)^(2)\mathbb{P}_{x, y}^{1} \times \mathbb{C}_{u, v}^{2}Px,y1×Cu,v2 via
(x:y;u,v)⟼(x:ζay;ζu,ζ−1v)(x:y;u,v)⟼x:ζay;ζu,ζ−1v(x:y;u,v)longmapsto(x:zeta^(a)y;zeta u,zeta^(-1)v)(x: y ; u, v) \longmapsto\left(x: \zeta^{a} y ; \zeta u, \zeta^{-1} v\right)(x:y;u,v)⟼(x:ζay;ζu,ζ−1v)
where ζ=ζn=exp(2πi/n)ζ=ζn=expâ¡(2Ï€i/n)zeta=zeta_(n)=exp(2pi i//n)\zeta=\zeta_{n}=\exp (2 \pi i / n)ζ=ζn=expâ¡(2Ï€i/n) and gcd(n,a)=1gcdâ¡(n,a)=1gcd(n,a)=1\operatorname{gcd}(n, a)=1gcdâ¡(n,a)=1. Then the projection
is a QQQ\mathbb{Q}Q-conic bundle. The variety XXXXX has exactly two singular points which are terminal cyclic quotients of type 1n(1,−1,±a)1n(1,−1,±a)(1)/(n)(1,-1,+-a)\frac{1}{n}(1,-1, \pm a)1n(1,−1,±a). The surface ZZZZZ has at 0 a DuDuDu\mathrm{Du}Du Val point of type An−1An−1A_(n-1)\mathrm{A}_{\mathrm{n}-1}An−1.
McKernan proposed a natural higher-dimensional analogue of Corollary 7.2:
Conjecture 7.4. Let f:X→Zf:X→Zf:X rarr Zf: X \rightarrow Zf:X→Z be a KKKKK-negative contraction such that ρ(X/Z)=1Ï(X/Z)=1rho(X//Z)=1\rho(X / Z)=1Ï(X/Z)=1 and XXXXX is εεepsi\varepsilonε-lc, that is, all the coefficients in (1.1.1) satisfy ai≥−1+εai≥−1+εa_(i) >= -1+epsia_{i} \geq-1+\varepsilonai≥−1+ε. Then ZZZZZ is δδdelta\deltaδ-lc, where δδdelta\deltaδ depends on εεepsi\varepsilonε and the dimension.
A deeper version of this conjecture which generalizes Theorem 5.1 and uses the notion was proposed by Shokurov. He also suggested that the optimal value of δδdelta\deltaδ, in the
case where singularities of XXXXX are canonical and fffff has one-dimensional fibers, equals 1/21/21//21 / 21/2. Recently, this was proved by J. Han, C. Jiang, and Y. Luo [17].
Once we have a Du Val general elephants, all extremal curve germs can be divided into two large classes which will be discussed separately in the next two sections.
Definition 7.5. Let (X⊃C)(X⊃C)(X sup C)(X \supset C)(X⊃C) be an extremal curve germ and let f:X→(Z∋o)f:X→(Z∋o)f:X rarr(Z∋o)f: X \rightarrow(Z \ni o)f:X→(Z∋o) be the corresponding contraction. Assume that the general member D∈|−KX|D∈−KXD in|-K_(X)|D \in\left|-K_{X}\right|D∈|−KX| is Du Val. Consider the Stein factorization:
fD:D⟶D′⟶f(D) (put D′=f(D) if f is birational). fD:D⟶D′⟶f(D) (put D′=f(D) if f is birational). f_(D):D longrightarrowD^(')longrightarrow f(D)quad" (put "D^(')=f(D)" if "f" is birational). "f_{D}: D \longrightarrow D^{\prime} \longrightarrow f(D) \quad \text { (put } D^{\prime}=f(D) \text { if } f \text { is birational). }fD:D⟶D′⟶f(D) (put D′=f(D) if f is birational).Â
Then the germ (X⊃C)(X⊃C)(X sup C)(X \supset C)(X⊃C) is said to be semistable if D′D′D^(')D^{\prime}D′ has only (Du Val) singularities of type AnAnA_(n)\mathrm{A}_{\mathrm{n}}An. Otherwise, (X⊃C)(X⊃C)(X sup C)(X \supset C)(X⊃C) is called exceptional.
8. SEMISTABLE GERMS
Let (X⊃C)(X⊃C)(X sup C)(X \supset C)(X⊃C) be an irreducible extremal curve germ. By Theorem 7.1, the general member D∈|−KX|D∈−KXD in|-K_(X)|D \in\left|-K_{X}\right|D∈|−KX| is Du Val. In this section we assume that (X⊃C)(X⊃C)(X sup C)(X \supset C)(X⊃C) is semistable. Excluding simple cases, we assume also that XXXXX is not Gorenstein [12] and (X⊃C)(X⊃C)(X sup C)(X \supset C)(X⊃C) is not a QQQ\mathbb{Q}Q-conic bundle germ over a singular base [47,48][47,48][47,48][47,48][47,48]. According to Theorem 6.4, the threefold XXXXX has at most two non-Gorenstein points. Thus the following case division is natural:
Case (k1A): the set of non-Gorenstein points consists of a single point PPPPP;
Case (k2A): the set of non-Gorenstein points consists of exactly two points P1,P2P1,P2P_(1),P_(2)P_{1}, P_{2}P1,P2.
Proposition 8.1. In the above hypothesis, for the general member H∈|OX|H∈OXH in|O_(X)|H \in\left|\mathscr{O}_{X}\right|H∈|OX|, the pair (X,H+D)(X,H+D)(X,H+D)(X, H+D)(X,H+D) is lc. If, moreover, D⊃CD⊃CD sup CD \supset CD⊃C, then HHHHH is normal and has only cyclic quotient singularities. In this case the singularities of HHHHH are of type TTT\mathrm{T}T.
The proof uses the inversion of adjunction [70] to extend a general hyperplane section from DDDDD to XXXXX (see [51, PROPOSITION 2.6]).
For an extremal curve germ of type (k2A), any member D∈|−KX|D∈−KXD in|-K_(X)|D \in\left|-K_{X}\right|D∈|−KX| contains CCCCC [38]. Hence the general hyperplane section H∈|OX|H∈OXH in|O_(X)|H \in\left|\mathscr{O}_{X}\right|H∈|OX| has only T-singularities and XXXXX can be restored as a one-parameter deformation space of HHHHH. In this case XXXXX has no singularities other than P1,P2P1,P2P_(1),P_(2)P_{1}, P_{2}P1,P2. Moreover, (X⊃C)(X⊃C)(X sup C)(X \supset C)(X⊃C) cannot be a QQQ\mathbb{Q}Q-conic bundle germ [47,50]. The birational germs of type (k2A) were completely described by Mori [46]. He gave an explicit algorithm for computing divisorial contractions and flips in this case.
The structure of extremal curve germs of type (k1A) is more complicated. They were studied in [51]. In particular, the general hyperplane section H∈|OX|H∈OXH in|O_(X)|H \in\left|\mathscr{O}_{X}\right|H∈|OX| was computed. However, [51] does not provide a good description of the infinitesimal structure of XXXXX along CCCCC or an algorithm similar to [46]. This was done only in a special situation in [14]. Note that in the case (k1A)(k1A)(k1A)(\mathrm{k} 1 \mathrm{~A})(k1 A) a general member H∈|OX|H∈OXH in|O_(X)|H \in\left|\mathscr{O}_{X}\right|H∈|OX| can be nonnormal.
Examples. Similar to the example in Section 6.1, consider a surface germ H⊃C≃P1H⊃C≃P1H sup C≃P^(1)H \supset C \simeq \mathbb{P}^{1}H⊃C≃P1 whose dual graph has the following graph of the minimal resolution:
where ∙∙∙\bullet∙ is a ( -1 -curve. The chain formed by white circle vertices o corresponds to a Tsingularity of type 125(1,4)125(1,4)(1)/(25)(1,4)\frac{1}{25}(1,4)125(1,4). The whole configuration can be contracted to a cyclic quotient singularity HZ∋oHZ∋oH_(Z)∋oH_{Z} \ni oHZ∋o of type 121(1,16)121(1,16)(1)/(21)(1,16)\frac{1}{21}(1,16)121(1,16). Since this is not a T-singularity, the induced threefold contraction must be flipping.
9. EXCEPTIONAL CURVE GERMS
In this section we assume that (X⊃C)(X⊃C)(X sup C)(X \supset C)(X⊃C) is an exceptional irreducible extremal curve germ. As in the previous section we also assume that XXXXX is not Gorenstein and (X⊃C)(X⊃C)(X sup C)(X \supset C)(X⊃C) is not a QQQ\mathbb{Q}Q-conic bundle germ over a singular base. According to the classification [38,44,50][38,44,50][38,44,50][38,44,50][38,44,50], the germ (X⊃C)(X⊃C)(X sup C)(X \supset C)(X⊃C) belongs to one of following types:
XXXXX has a unique non-Gorenstein point PPPPP which is of type cD/2, cAx/2, cE/2, or cD/3cD/3cD//3\mathrm{cD} / 3cD/3 and (X⊃C)(X⊃C)(X sup C)(X \supset C)(X⊃C) is of type (IA) at PPPPP;
XXXXX has a unique non-Gorenstein point PPPPP which is of exceptional type cAx/4cAx/4cAx//4\mathrm{cAx} / 4cAx/4 and (X⊃C)(X⊃C)(X sup C)(X \supset C)(X⊃C) is of type (IIA), (II ∨)∨{:^(vv))\left.{ }^{\vee}\right)∨), or (IIB) at PPPPP;
XXXXX has a unique singular point PPPPP which is a cyclic quotient singularity of index m≥5m≥5m >= 5m \geq 5m≥5 (odd) and (X⊃C)(X⊃C)(X sup C)(X \supset C)(X⊃C) is of type (IC) at PPPPP;
XXXXX has two singular points of indices m≥3m≥3m >= 3m \geq 3m≥3 (odd) and 2, then (X⊃C)(X⊃C)(X sup C)(X \supset C)(X⊃C) is said to be of type (kAD);
XXXXX has three singular points of indices m≥3m≥3m >= 3m \geq 3m≥3 (odd), 2 and 1 , then (X⊃C)(X⊃C)(X sup C)(X \supset C)(X⊃C) is said to be of type (k3A)(k3A)(k3A)(\mathrm{k} 3 \mathrm{~A})(k3 A).
In each case the general elephant is completely described in terms of its minimal resolution:
Theorem 9.1 ([38,50]). In the above hypothesis assume that the general element D∈|−KX|D∈−KXD in|-K_(X)|D \in\left|-K_{X}\right|D∈|−KX| contains CCCCC. Then the dual graph of (D⊃C)(D⊃C)(D sup C)(D \supset C)(D⊃C) is one of the following, where white vertices ∘∘@\circ∘ denote (-2)-curves on the minimal resolution of DDDDD and the black vertex ∙∙∙\bullet∙ corresponds to the proper transform of CCCCC :
Exceptional irreducible extremal curve germs are well studied (see [38,55][38,55][38,55][38,55][38,55], and references therein). For flipping ones, the general hyperplane section H∈|OX|H∈OXH in|O_(X)|H \in\left|\mathscr{O}_{X}\right|H∈|OX| is normal and has only rational singularities. It is computed in [38] and the flip is reconstructed as a oneparameter deformation space of HHHHH. For divisorial and QQQ\mathbb{Q}Q-conic bundle germs, the situation is more complicated. Then the general hyperplane section HHHHH can be nonnormal (see, e.g., [54]). Nevertheless, in almost all cases, except for types (kAD) and (k3A), there is a description of H∈|OX|H∈OXH in|O_(X)|H \in\left|\mathscr{O}_{X}\right|H∈|OX| and infinitesimal structure of these germs. For convenience of reference, in the table below we collect the known information on the exceptional irreducible extremal curve germs.
Detailed analysis of the local structure of exceptional extremal curve germs allows extending the result of Theorem 7.1 to the case of reducible central fiber containing an exceptional component:
Theorem 9.2 (Mori-Prokhorov [56]). Let (X⊃C)(X⊃C)(X sup C)(X \supset C)(X⊃C) be an extremal curve germ such that CCCCC is reducible and satisfies the following condition:
(∗)(∗)^((**)){ }^{(*)}(∗) each component Ci⊂CCi⊂CC_(i)sub CC_{i} \subset CCi⊂C contains at most one point of index >2>2> 2>2>2.
Then the general member D∈|−KX|D∈−KXD in|-K_(X)|D \in\left|-K_{X}\right|D∈|−KX| has only Du Val singularities. Moreover, for each irreducible component Ci⊂CCi⊂CC_(i)sub CC_{i} \subset CCi⊂C with two non-Gorenstein points or of types (IC) or (IIB), the dual graph of (D,Ci)D,Ci(D,C_(i))\left(D, C_{i}\right)(D,Ci) has the same form as the irreducible extremal curve germ (X⊃Ci)X⊃Ci(X supC_(i))\left(X \supset C_{i}\right)(X⊃Ci).
The proof uses the extension techniques of sections of |−KX|−KX|-K_(X)|\left|-K_{X}\right||−KX| from a good member S∈|−2KX|S∈−2KXS in|-2K_(X)|S \in\left|-2 K_{X}\right|S∈|−2KX| (see Section 7.2).
where mPmPm_(P)m_{P}mP is the index of a virtual quotient singularity of XXXXX [66]. Since XXXXX is QQQ\mathbb{Q}Q-Fano, by Kawamata-Viehweg vanishing theorem [35], one has χ(OX)=1χOX=1chi(O_(X))=1\chi\left(\mathscr{O}_{X}\right)=1χ(OX)=1. Arguments based on Bogomolov-Miyaoka inequality show that −KX⋅c2(X)−KXâ‹…c2(X)-K_(X)*c_(2)(X)-K_{X} \cdot \mathrm{c}_{2}(X)−KXâ‹…c2(X) is positive (see [33]). This gives an effective bound of the indices of singularities of XXXXX. Similarly, one can get an upper bound of the anticanonical degree −KX3−KX3-K_(X)^(3)-K_{X}^{3}−KX3. Moreover, analyzing the methods of [33], it is possible to enumerate Hilbert series of all QQQ\mathbb{Q}Q-Fano threefolds. This information is collected in [6] in a form of a huge computer database of possible "candidates." It was extensively explored by many authors, basically to obtain lists of examples representing QQQ\mathbb{Q}Q-Fano threefolds as subvarieties of small codimension in a weighted projective space (see, e.g., [7,21][7,21][7,21][7,21][7,21] and references therein).
Examples. - There are 130 (resp. 125) families of QQQ\mathbb{Q}Q-Fano threefolds that are representable as hypersurfaces (resp. codimension 2 complete intersections) in weighted projective spaces [6,21][6,21][6,21][6,21][6,21].
Toric QQQ\mathbb{Q}Q-Fano threefolds are exactly weighted projective spaces P(3,4,5,7)P(3,4,5,7)P(3,4,5,7)\mathbb{P}(3,4,5,7)P(3,4,5,7), P(2,3,5,7),P(1,3,4,5),P(1,2,3,5),P(1,1,2,3),P(1,1,1,2),P3=P(1,1P(2,3,5,7),P(1,3,4,5),P(1,2,3,5),P(1,1,2,3),P(1,1,1,2),P3=P(1,1P(2,3,5,7),P(1,3,4,5),P(1,2,3,5),P(1,1,2,3),P(1,1,1,2),P^(3)=P(1,1\mathbb{P}(2,3,5,7), \mathbb{P}(1,3,4,5), \mathbb{P}(1,2,3,5), \mathbb{P}(1,1,2,3), \mathbb{P}(1,1,1,2), \mathbb{P}^{3}=\mathbb{P}(1,1P(2,3,5,7),P(1,3,4,5),P(1,2,3,5),P(1,1,2,3),P(1,1,1,2),P3=P(1,1, 1,1)1,1)1,1)1,1)1,1), and the quotient of P3P3P^(3)\mathbb{P}^{3}P3 by μ5μ5mu_(5)\boldsymbol{\mu}_{5}μ5 that acts diagonally with weights (1,2,3,4)[6](1,2,3,4)[6](1,2,3,4)[6](1,2,3,4)[6](1,2,3,4)[6].
Although the classification is very far from completion, there are several systematic results. For example, the optimal upper bound of the degree −KX3−KX3-K_(X)^(3)-K_{X}^{3}−KX3 of QQQ\mathbb{Q}Q-Fano threefolds was obtained in [58]. If XXXXX is singular, it is equal to 125/2 and achieved for the weighted projective space P(1,1,1,2)P(1,1,1,2)P(1,1,1,2)\mathbb{P}(1,1,1,2)P(1,1,1,2). The lower bound of the degree equals 1/3301/3301//3301 / 3301/330 [8] and is achieved for a hypersurface of degree 66 in P(1,5,6,22,33)P(1,5,6,22,33)P(1,5,6,22,33)\mathbb{P}(1,5,6,22,33)P(1,5,6,22,33). It is known that, under certain conditions, General Elephant Conjecture 3.1 holds for QQQ\mathbb{Q}Q-Fano threefolds modulo deformations [67].
ACKNOWLEDGMENTS
The author would like to thank Professors Shigefumi Mori and Vyacheslav Shokurov for helpful comments on the original version of this paper.
FUNDING
This work was performed at the Steklov International Mathematical Center and supported by the Ministry of Science and Higher Education of the Russian Federation (agreement no. 075-15-2019-1614).
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SOME ASPECTS OF RATIONAL POINTS AND RATIONAL CURVES
OLIVIER WITTENBERG
ABSTRACT
Various methods have been used to construct rational points and rational curves on rationally connected algebraic varieties. We survey recent advances in two of them, the descent and the fibration method, in a number-theoretical context (rational points over number fields) and in an algebro-geometric one (rational curves on real varieties), and discuss the question of rational points over function fields of ppppp-adic curves.
Criteria for the existence of rational points on XXXXX are also of relevance outside of number theory, when kkkkk is no longer assumed to be a number field. For instance, the GraberHarris-Starr theorem [34], a central result in the theory of rational curves on complex algebraic varieties, is equivalent to the statement that X(k)≠∅X(k)≠∅X(k)!=O/X(k) \neq \varnothingX(k)≠∅ if kkkkk is the function field of a complex curve and XXXXX is a rationally connected variety. (We say that XXXXX is rationally connected to mean that for any algebraically closed field extension KKKKK of kkkkk, the variety XKXKX_(K)X_{K}XK over KKKKK is rationally connected in the sense of [12,69][12,69][12,69][12,69][12,69].) As another example, if XXXXX is a real algebraic variety with no real point and kkkkk denotes the function field of the real conic given by x2+y2=−1x2+y2=−1x^(2)+y^(2)=-1x^{2}+y^{2}=-1x2+y2=−1, the existence of a geometrically rational curve on XXXXX-a property conjectured by Kollár to hold whenever XXXXX is a positive-dimensional rationally connected variety-is equivalent to the statement that X(k)≠∅X(k)≠∅X(k)!=O/X(k) \neq \varnothingX(k)≠∅.
The results we discuss in this expository article concern the existence of rational points in two very distinct contexts, leading to the following two concrete theorems, obtained in collaboration with Yonatan Harpaz and with Olivier Benoist, respectively. As we shall see, their proofs roughly follow, perhaps somewhat surprisingly, a common general strategy.
Theorem A (see [50]). Let GGGGG be a finite nilpotent group. Let kkkkk be a number field.
(1) There exist Galois extensions K/kK/kK//kK / kK/k with Galois group GGGGG.
(2) If v1,…,vnv1,…,vnv_(1),dots,v_(n)v_{1}, \ldots, v_{n}v1,…,vn are pairwise distinct places of kkkkk none of which is a finite place dividing the order of GGGGG, and w1,…,wnw1,…,wnw_(1),dots,w_(n)w_{1}, \ldots, w_{n}w1,…,wn are places of KKKKK above v1,…,vnv1,…,vnv_(1),dots,v_(n)v_{1}, \ldots, v_{n}v1,…,vn, then in (1), one can require that the extensions Kwi/kviKwi/kviK_(w_(i))//k_(v_(i))K_{w_{i}} / k_{v_{i}}Kwi/kvi be isomorphic to any prescribed collection of Galois extensions of kv1,…,kvnkv1,…,kvnk_(v_(1)),dots,k_(v_(n))k_{v_{1}}, \ldots, k_{v_{n}}kv1,…,kvn whose Galois groups are subgroups of GGGGG.
Theorem B (see [6]). Let XXXXX be a smooth, proper variety over RRR\mathbf{R}R. Let ε:S1→X(R)ε:S1→X(R)epsi:S^(1)rarr X(R)\varepsilon: \mathbf{S}^{1} \rightarrow X(\mathbf{R})ε:S1→X(R) be a continuous map. Assume that XXXXX is birationally equivalent to a homogeneous space of a linear algebraic group over RRR\mathbf{R}R. Then there exist morphisms of algebraic varieties PR1→XPR1→XP_(R)^(1)rarr X\mathbf{P}_{\mathbf{R}}^{1} \rightarrow XPR1→X that induce maps P1(R)=S1→X(R)P1(R)=S1→X(R)P^(1)(R)=S^(1)rarr X(R)\mathbf{P}^{1}(\mathbf{R})=\mathbf{S}^{1} \rightarrow X(\mathbf{R})P1(R)=S1→X(R) arbitrarily close to εεepsi\varepsilonε in the compact-open topology.
Theorem A (1) was first proved by Shafarevich in his seminal work on the inverse Galois problem for solvable groups (see [82, CHAPTER IX, § 6]; it should be noted that nilpotent groups form the most difficult case in his proof); the proof given in [50] is independent from his and has a geometric flavour. Theorem A (2), on the other hand, was new in [50] and was not accessible with Shafarevich's methods.
As far as we know, Theorem B might hold under the sole assumption that XXXXX is rationally connected. This is a question we raise in [6]. Theorem B provides the first examples of a positive answer to it for varieties that are not RRR\mathbf{R}R-rational (indeed, not even CCC\mathbf{C}C-rational). For RRR\mathbf{R}R-rational varieties, the conclusion of Theorem B was previously shown, by Bochnak and Kucharz [8], to follow from the Stone-Weierstrass theorem.
We take Theorems A and B as excuses leading us to the general study of rational points on rationally connected varieties defined over number fields or over function fields of real curves. We discuss recent advances in the fibration and descent methods in these two contexts in Sections 2 and 3, stating along the way the main open questions that surround Theorems A and B and their proofs. We then turn, in Section 4, to function fields of ppppp-adic curves, and speculate about the existence of a ppppp-adic analogue of the "tight approximation"
property discussed in Section 3 that would enable one to exploit fibration and descent methods in the study of rational curves over ppppp-adic fields and more generally of rational points over function fields of ppppp-adic curves.
2. SOLVABLE GROUPS AND THE GRUNWALD PROBLEM IN INVERSE GALOIS THEORY
2.1. Homogeneous spaces
It is the following general theorem about the arithmetic of homogeneous spaces of linear algebraic groups that underlies Theorem A.
Theorem 2.1. Let VVVVV be a homogeneous space of a connected linear algebraic group LLLLL over a number field kkkkk. Let XXXXX be a smooth compactification of VVVVV. Let v¯∈V(k¯)v¯∈V(k¯)bar(v)in V( bar(k))\bar{v} \in V(\bar{k})v¯∈V(k¯). Assume that the group of connected components GGGGG of the stabiliser of v¯v¯bar(v)\bar{v}v¯ is supersolvable, in the sense that it possesses a normal series 1=G0◃⋯◃Gm=G1=G0◃⋯◃Gm=G1=G_(0)â—ƒcdotsâ—ƒG_(m)=G1=G_{0} \triangleleft \cdots \triangleleft G_{m}=G1=G0◃⋯◃Gm=G such that the quotients Gi+1/GiGi+1/GiG_(i+1)//G_(i)G_{i+1} / G_{i}Gi+1/Gi are cyclic while the subgroups GiGiG_(i)G_{i}Gi are normal in GGGGG and are stable under the natural outer action of Gal(k¯/k)Galâ¡(k¯/k)Gal( bar(k)//k)\operatorname{Gal}(\bar{k} / k)Galâ¡(k¯/k) on GGGGG. Then the subset X(k)X(k)X(k)X(k)X(k) is dense in X(Ak)Br(X)XAkBrâ¡(X)X(A_(k))^(Br(X))X\left(\mathbf{A}_{k}\right)^{\operatorname{Br}(X)}X(Ak)Brâ¡(X).
Here and elsewhere, by "compactification of VVVVV," we mean a proper variety over kkkkk that contains VVVVV as a dense open subset; we do not require that the algebraic group LLLLL act on the compactification. Examples of supersolvable groups with respect to the trivial outer action of Gal(k¯/k)Galâ¡(k¯/k)Gal( bar(k)//k)\operatorname{Gal}(\bar{k} / k)Galâ¡(k¯/k) include finite nilpotent groups and dihedral groups. With a nontrivial outer action of Gal(k¯/k)Galâ¡(k¯/k)Gal( bar(k)//k)\operatorname{Gal}(\bar{k} / k)Galâ¡(k¯/k), however, even abelian groups need not be supersolvable. Previous work of Borovoi [10] nevertheless establishes the conclusion of Theorem 2.1 in many cases where the stabiliser of v¯v¯bar(v)\bar{v}v¯ is abelian but not necessarily supersolvable.
X(Ak)Br(X)XAkBrâ¡(X)X(A_(k))^(Br(X))X\left(\mathbf{A}_{k}\right)^{\operatorname{Br}(X)}X(Ak)Brâ¡(X); that is, one may freely prescribe the completions of KKKKK at any finite set of places of kkkkk, as long as these prescriptions satisfy a certain global reciprocity condition determined by Br(X)Brâ¡(X)Br(X)\operatorname{Br}(X)Brâ¡(X). By a theorem of Lucchini Arteche [75, 86 6, this reciprocity condition imposes, in fact, no restriction at the places indicated in Theorem A (2).
This geometry is the key to a proof of Theorem 2.1 by an induction on mmmmm, at each step of which one applies the descent method and the fibration method, in the form of Theorems 2.2 and 2.3 below. It should be noted that even if GGGGG is embedded into SLn(k)SLn(k)SL_(n)(k)\mathrm{SL}_{n}(k)SLn(k) and V=SLn/GV=SLn/GV=SL_(n)//GV=\mathrm{SL}_{n} / GV=SLn/G, the homogeneous spaces of SLnSLnSL_(n)\mathrm{SL}_{n}SLn that arise as fibres of ppppp need not possess rational points. Thus, for the induction to be possible, one cannot restrict the statement of Theorem 2.1 to homogeneous spaces of the form SLn/GSLn/GSL_(n)//G\mathrm{SL}_{n} / GSLn/G, even though only homogeneous spaces of this form are relevant for Theorem A.
Theorem 2.2. Let XXXXX be a smooth and proper rationally connected variety over a number field kkkkk. Let TTTTT be a torus over kkkkk and Y¯→Xk¯Y¯→Xk¯bar(Y)rarrX_( bar(k))\bar{Y} \rightarrow X_{\bar{k}}Y¯→Xk¯ a torsor under Tk¯Tk¯T_( bar(k))T_{\bar{k}}Tk¯ whose isomorphism class is invariant under Gal(k¯/k)Galâ¡(k¯/k)Gal( bar(k)//k)\operatorname{Gal}(\bar{k} / k)Galâ¡(k¯/k). Then
X(Ak)Br(X)=⋃f:Y→Xf′(Y′(Ak)Br(Y′))XAkBrâ¡(X)=⋃f:Y→X f′Y′AkBrâ¡Y′X(A_(k))^(Br(X))=uuu_(f:Y rarr X)f^(')(Y^(')(A_(k))^(Br(Y^('))))X\left(\mathbf{A}_{k}\right)^{\operatorname{Br}(X)}=\bigcup_{f: Y \rightarrow X} f^{\prime}\left(Y^{\prime}\left(\mathbf{A}_{k}\right)^{\operatorname{Br}\left(Y^{\prime}\right)}\right)X(Ak)Brâ¡(X)=⋃f:Y→Xf′(Y′(Ak)Brâ¡(Y′))
where the union ranges over the torsors f:Y→Xf:Y→Xf:Y rarr Xf: Y \rightarrow Xf:Y→X under TTTTT whose base change to Xk¯Xk¯X_( bar(k))X_{\bar{k}}Xk¯ is isomorphic to Y¯Y¯bar(Y)\bar{Y}Y¯, and Y′Y′Y^(')Y^{\prime}Y′ denotes a smooth compactification of YYYYY such that fffff extends to a morphism f′:Y′→Xf′:Y′→Xf^('):Y^(')rarr Xf^{\prime}: Y^{\prime} \rightarrow Xf′:Y′→X. In particular, if Y′(k)Y′(k)Y^(')(k)Y^{\prime}(k)Y′(k) is dense in Y′(Ak)Br(Y′)Y′AkBrâ¡Y′Y^(')(A_(k))^(Br(Y^(')))Y^{\prime}\left(\mathbf{A}_{k}\right)^{\operatorname{Br}\left(Y^{\prime}\right)}Y′(Ak)Brâ¡(Y′) for every such fffff, then X(k)X(k)X(k)X(k)X(k) is dense in X(Ak)Br(X)XAkBrâ¡(X)X(A_(k))^(Br(X))X\left(\mathbf{A}_{k}\right)^{\operatorname{Br}(X)}X(Ak)Brâ¡(X).
(To bridge the gap between Theorem 2.2 and [50, THÉORÈME 2.1], one needs to know that X(Ak)Br(X)≠∅XAkBrâ¡(X)≠∅X(A_(k))^(Br(X))!=O/X\left(\mathbf{A}_{k}\right)^{\operatorname{Br}(X)} \neq \varnothingX(Ak)Brâ¡(X)≠∅ implies the existence of at least one fffff. This goes back to [25] and follows from [102, THEOREM 3.3.1], [25, PROPOSITION 2.2.5], [103, (3.3)].)
2.4. Fibration
The following fibration theorem suffices for the proof of Theorem 2.1. It results from combining a descent with the work of Harari [35] on the fibration method.
Theorem 2.3. Let p:Z→Bp:Z→Bp:Z rarr Bp: Z \rightarrow Bp:Z→B be a dominant morphism between irreducible, smooth, and proper varieties over a number field kkkkk, with rationally connected generic fibre. Assume that
(1) there exist dense open subsets W⊂ZW⊂ZW sub ZW \subset ZW⊂Z and Q⊂BQ⊂BQ sub BQ \subset BQ⊂B such that QQQQQ is a quasitrivial torus over kkkkk and ppppp induces a smooth morphism W→QW→QW rarr QW \rightarrow QW→Q with geometrically irreducible fibres;
(2) the morphism ppppp admits a rational section over k¯k¯bar(k)\bar{k}k¯;
(3) for all b∈B(k)b∈B(k)b in B(k)b \in B(k)b∈B(k) in a dense open subset of BBBBB, the set Zb(k)Zb(k)Z_(b)(k)Z_{b}(k)Zb(k) is dense in Zb(Ak)Br(Zb)ZbAkBrâ¡ZbZ_(b)(A_(k))^(Br(Z_(b)))Z_{b}\left(\mathbf{A}_{k}\right)^{\operatorname{Br}\left(Z_{b}\right)}Zb(Ak)Brâ¡(Zb)
Then Z(k)Z(k)Z(k)Z(k)Z(k) is dense in Z(Ak)Br(Z)ZAkBrâ¡(Z)Z(A_(k))^(Br(Z))Z\left(\mathbf{A}_{k}\right)^{\operatorname{Br}(Z)}Z(Ak)Brâ¡(Z).
The assumptions of Theorem 2.3 imply that BBBBB is kkkkk-rational. Under the condition that BBBBB is kkkkk-rational, the first two assumptions of Theorem 2.3 are expected to be superfluous (even under weaker hypotheses on the generic fibre of ppppp than rational connectedness, see [48, COROLLARY 9.23 (1)-(2)]), but removing them altogether is a wide-open problem, well connected with analytic number theory (see [48, § 9], [47]). Removing (2) while keeping (1) might be within reach, though:
Question 2.4. In the statement of Theorem 2.3, can one dispense with the assumption that ppppp admit a rational section over k¯k¯bar(k)\bar{k}k¯ ?
This would allow one to replace "supersolvable" with "solvable" in the statement of Theorem 2.1. Indeed, in Section 2.2, the cyclicity of the quotient Gm/Gm−1Gm/Gm−1G_(m)//G_(m-1)G_{m} / G_{m-1}Gm/Gm−1 plays a rôle only to ensure the existence of a rational section of ppppp over k¯k¯bar(k)\bar{k}k¯ (see [50, PROPOSITION 3.3 (II)]).
2.5. An application to Massey products
Theorem 2.1 has concrete applications, over number fields, beyond the inverse Galois problem: for the homogeneous spaces that appear in its statement, it turns the problem of deciding the existence of a rational point into the much more approachable question of deciding the non-vacuity of the Brauer-Manin set. In this way, Theorem 2.1 can be used to confirm, in the case of number fields, the conjecture of MinÃ¡Ä and Tân on the vanishing of Massey products in Galois cohomology (see [49]). Indeed, this conjecturewhich posits that for any field kkkkk, any prime number ppppp, any integer m≥3m≥3m >= 3m \geq 3m≥3 and any classes a1,…,am∈H1(k,Z/pZ)a1,…,am∈H1(k,Z/pZ)a_(1),dots,a_(m)inH^(1)(k,Z//pZ)a_{1}, \ldots, a_{m} \in H^{1}(k, \mathbf{Z} / p \mathbf{Z})a1,…,am∈H1(k,Z/pZ), the mmmmm-fold Massey product of a1,…,ama1,…,ama_(1),dots,a_(m)a_{1}, \ldots, a_{m}a1,…,am vanishes if it is defined (see [78,79])—can be reinterpreted, according to Pál and Schlank [83], in terms of the existence of rational points on appropriate homogeneous spaces of SLnSLnSL_(n)\mathrm{SL}_{n}SLn over kkkkk (with n≫0n≫0n≫0n \gg 0n≫0 ), and it so happens that the geometric stabilisers of these homogeneous spaces are finite and supersolvable.
3. RATIONAL CURVES ON REAL ALGEBRAIC VARIETIES
3.1. A few questions
Let XXXXX be a smooth variety over RRR\mathbf{R}R. The interplay between the topology of the C∞C∞C^(oo)\mathscr{C}^{\infty}C∞ manifold X(R)X(R)X(R)X(\mathbf{R})X(R) and the geometry of the algebraic variety XXXXX lies at the core of classical
real algebraic geometry. One of the fundamental problems in this area consists in investigating which submanifolds of X(R)X(R)X(R)X(\mathbf{R})X(R) can be approximated, in the Euclidean topology, by Zariski closed submanifolds. Even for 1-dimensional submanifolds, i.e., disjoint unions of C∞C∞C^(oo)\mathscr{C}^{\infty}C∞ loops, various phenomena-of a topological, Hodge-theoretic, or yet more subtle nature-can obstruct the existence of algebraic approximations (see [4, § 4]). In the case of 1-dimensional submanifolds, however, all known obstructions vanish when XXXXX is rationally connected. One can thus raise the following questions, in which H1alg (X(R),Z/2Z)H1alg (X(R),Z/2Z)H_(1)^("alg ")(X(R),Z//2Z)H_{1}^{\text {alg }}(X(\mathbf{R}), \mathbf{Z} / 2 \mathbf{Z})H1alg (X(R),Z/2Z) denotes the image of the cycle class map CH1(X)→H1(X(R),Z/2Z)CH1(X)→H1(X(R),Z/2Z)CH_(1)(X)rarrH_(1)(X(R),Z//2Z)\mathrm{CH}_{1}(X) \rightarrow H_{1}(X(\mathbf{R}), \mathbf{Z} / 2 \mathbf{Z})CH1(X)→H1(X(R),Z/2Z) defined by Borel and Haefliger [9].
Questions 3.1. Let XXXXX be a smooth, proper, rationally connected variety, over RRR\mathbf{R}R.
(1) Can all C∞C∞C^(oo)\mathscr{C}^{\infty}C∞ loops in X(R)X(R)X(R)X(\mathbf{R})X(R) be approximated, in the Euclidean topology, by real loci of algebraic curves? or even by real loci of rational algebraic curves?
(2) Is H1(X(R),Z/2Z)=H1alg (X(R),Z/2Z)H1(X(R),Z/2Z)=H1alg (X(R),Z/2Z)H_(1)(X(R),Z//2Z)=H_(1)^("alg ")(X(R),Z//2Z)H_{1}(X(\mathbf{R}), \mathbf{Z} / 2 \mathbf{Z})=H_{1}^{\text {alg }}(X(\mathbf{R}), \mathbf{Z} / 2 \mathbf{Z})H1(X(R),Z/2Z)=H1alg (X(R),Z/2Z) ? Is H1(X(R),Z/2Z)H1(X(R),Z/2Z)H_(1)(X(R),Z//2Z)H_{1}(X(\mathbf{R}), \mathbf{Z} / 2 \mathbf{Z})H1(X(R),Z/2Z) generated by classes of rational algebraic curves on XXXXX ?
The first parts of Questions 3.1 (1) and (2) are in fact equivalent to each other, by the work of Akbulut and King (see [5, Ñ‚HEOREM 6.8]), and were studied in a systematic fashion in [4,5]. The second part of Question 3.1 (1) is, however, as far as we know, genuinely stronger than the second part of Question 3.1 (2). We note that in order to formulate the second part of Question 3.1 (1) precisely, it is better to work with possibly noninjective C∞C∞C^(oo)\mathscr{C}^{\infty}C∞ maps P1(R)→X(R)P1(R)→X(R)P^(1)(R)rarr X(R)\mathbf{P}^{1}(\mathbf{R}) \rightarrow X(\mathbf{R})P1(R)→X(R) rather than with submanifolds of X(R)X(R)X(R)X(\mathbf{R})X(R). Indeed, there are examples of RRR\mathbf{R}R-rational surfaces XXXXX and of C∞C∞C^(oo)\mathscr{C}^{\infty}C∞ loops in X(R)X(R)X(R)X(\mathbf{R})X(R) such that the desired rational algebraic curves necessarily have singular real points (see [68, THEOREM 3]).
A specific motivation for Question 3.1 (2) is its analogy with the following questions in complex geometry raised by Voisin [101] and by Kollár [67]:
Questions 3.2. Let XXXXX be a smooth, proper, rationally connected variety, over CCC\mathbf{C}C. Is the group H2(X(C),Z)H2(X(C),Z)H_(2)(X(C),Z)\mathrm{H}_{2}(\mathrm{X}(\mathbf{C}), \mathbf{Z})H2(X(C),Z) generated by homology classes of algebraic curves? Is it generated by homology classes of rational algebraic curves?
The two parts of Questions 3.2 are in fact equivalent: Tian and Zong [100] have shown that the homology class of any algebraic curve on a rationally connected variety over CCC\mathbf{C}C is a linear combination of homology classes of rational curves. The real analogue of their result remains unknown in general. Its validity is an interesting open problem.
The first parts of Questions 3.1 (2) and of Questions 3.2 are in fact related by more than an analogy: if XXXXX is a smooth, proper, rationally connected variety over RRR\mathbf{R}R such that X(R)≠∅X(R)≠∅X(R)!=O/X(\mathbf{R}) \neq \varnothingX(R)≠∅ and such that Questions 3.2 admit a positive answer for XCXCX_(C)X_{\mathbf{C}}XC, then the equality H1(X(R),Z/2Z)=H1alg (X(R),Z/2Z)H1(X(R),Z/2Z)=H1alg (X(R),Z/2Z)H_(1)(X(R),Z//2Z)=H_(1)^("alg ")(X(R),Z//2Z)H_{1}(X(\mathbf{R}), \mathbf{Z} / 2 \mathbf{Z})=H_{1}^{\text {alg }}(X(\mathbf{R}), \mathbf{Z} / 2 \mathbf{Z})H1(X(R),Z/2Z)=H1alg (X(R),Z/2Z) is equivalent to the real integral Hodge conjecture for 1-cycles on XXXXX, a property formulated and studied in [4,5].
In a different line of investigation around the abundance of rational curves on rationally connected varieties, many authors have considered the problem of finding rational curves through a prescribed set of points, or more generally through a prescribed curvilinear
0-dimensional subscheme, on any smooth, proper, rationally connected variety XXXXX. Over the complex numbers, such curves exist unconditionally (Kollár, Miyaoka, Mori, see [63, chAPTER IV.3]). Over the real numbers, such curves exist under the necessary condition that all the prescribed points that are real belong to the same connected component of X(R)X(R)X(R)X(\mathbf{R})X(R) (Kollár, see [64,66])[64,66])[64,66])[64,66])[64,66]). This problem can be generalised to one-parameter families: given a morphism f:X→Bf:X→Bf:Xrarr Bf: \mathscr{X} \rightarrow Bf:X→B with rationally connected generic fibre between smooth and proper varieties, where BBBBB is a curve, one looks for sections of fffff whose restriction to a given 0 -dimensional subscheme of BBBBB is prescribed, thus leading to Questions 3.3 below. For simplicity of notation, in the statement of Question 3.3 (2), this 0 -dimensional subscheme of BBBBB is assumed to be reduced; there is, however, no loss of generality in doing this, since jets of sections can be prescribed at any higher order by replacing XXX\mathscr{X}X with a suitable iterated blow-up (see [52, PROPOSITION 1.4]).
Questions 3.3. Let BBBBB be a smooth, proper, connected curve over a field k0k0k_(0)k_{0}k0. Let XXX\mathscr{X}X be a smooth, proper variety over k0k0k_(0)k_{0}k0, endowed with a flat morphism f:X→Bf:X→Bf:Xrarr Bf: \mathscr{X} \rightarrow Bf:X→B with rationally connected generic fibre. Let P⊂BP⊂BP sub BP \subset BP⊂B be a reduced 0-dimensional subscheme. Let s:P→Xs:P→Xs:P rarrXs: P \rightarrow \mathscr{X}s:P→X be a section of fffff over PPPPP.
(1) If k0=Ck0=Ck_(0)=Ck_{0}=\mathbf{C}k0=C, can sssss be extended to a section of fffff ?
(2) If k0=Rk0=Rk_(0)=Rk_{0}=\mathbf{R}k0=R and the map s|P(R):P(R)→X(R)sP(R):P(R)→X(R)s|_(P(R)):P(R)rarrX(R)\left.s\right|_{P(\mathbf{R})}: P(\mathbf{R}) \rightarrow \mathscr{X}(\mathbf{R})s|P(R):P(R)→X(R) can be extended to a C∞C∞C^(oo)\mathscr{C}^{\infty}C∞ section of f|X(R):X(R)→B(R)fX(R):X(R)→B(R)f|_(X(R)):X(R)rarr B(R)\left.f\right|_{\mathscr{X}(\mathbf{R})}: \mathscr{X}(\mathbf{R}) \rightarrow B(\mathbf{R})f|X(R):X(R)→B(R), can then sssss be extended to a section of fffff ?
Let XXXXX be the generic fibre of fffff and kkkkk the function field of BBBBB. The existence of sections extending any given sssss as above is equivalent to the density of X(k)X(k)X(k)X(k)X(k) in the topological space X(Ak)=∏bX(kb)XAk=âˆb XkbX(A_(k))=prod_(b)X(k_(b))X\left(\mathbf{A}_{k}\right)=\prod_{b} X\left(k_{b}\right)X(Ak)=âˆbX(kb) of adelic points of XXXXX, where the product runs over the closed points bbbbb of BBBBB and kbkbk_(b)k_{b}kb denotes the completion of kkkkk at bbbbb. This is the weak approximation property.
The main insight behind the proof of Theorem B is the observation that formulating a suitable common strengthening of Questions 3.1 and 3.3, through the notion of tight approximation, can render all of these questions fully amenable to both the descent method
and the fibration method. We note that Questions 3.1 and Questions 3.3 are somewhat orthogonal in spirit, insofar as the former consider global constraints on curves lying on XXXXX, while the latter are aimed at local constraints.
The idea of establishing a descent method (resp. fibration method) for Question 3.3 (2) already appeared in [31] (resp. [84]), though in [31] and [84] the implementations are subject to miscellaneous restrictions. The possibility of a descent method and a fibration method for studying Questions 3.1, however, is new and turns out to require a shift in perspective from single rationally connected varieties to one-parameter families of such.
Let us illustrate how Questions 3.1 need to be strengthened for a fibration argument to go through. We start with a dominant morphism p:X→Yp:X→Yp:X rarr Yp: X \rightarrow Yp:X→Y with rationally connected generic fibre between smooth, proper, rationally connected varieties, over RRR\mathbf{R}R, and a C∞C∞C^(oo)\mathscr{C}{ }^{\infty}C∞ loop γ:S1→X(R)γ:S1→X(R)gamma:S^(1)rarr X(R)\gamma: \mathbf{S}^{1} \rightarrow X(\mathbf{R})γ:S1→X(R) that we want to approximate, in the Euclidean topology, by a Zariski closed submanifold of XXXXX, assuming that we can solve the same problem on YYYYY as well as on the fibres of ppppp. By assumption, we can approximate p∘γ:S1→Y(R)p∘γ:S1→Y(R)p@gamma:S^(1)rarr Y(R)p \circ \gamma: \mathbf{S}^{1} \rightarrow Y(\mathbf{R})p∘γ:S1→Y(R) by a C∞C∞C^(oo)\mathscr{C}^{\infty}C∞ map ξ:S1→Y(R)ξ:S1→Y(R)xi:S^(1)rarr Y(R)\xi: \mathbf{S}^{1} \rightarrow Y(\mathbf{R})ξ:S1→Y(R) with Zariski closed image. The best we can hope to find, then, is a C∞C∞C^(oo)\mathscr{C}{ }^{\infty}C∞ loop γ~:S1→X(R)γ~:S1→X(R)tilde(gamma):S^(1)rarr X(R)\tilde{\gamma}: \mathbf{S}^{1} \rightarrow X(\mathbf{R})γ~:S1→X(R) arbitrarily close to γγgamma\gammaγ and such that p∘γ~=ξp∘γ~=ξp@ tilde(gamma)=xip \circ \tilde{\gamma}=\xip∘γ~=ξ. We draw two conclusions:
(1) If such a γ~γ~tilde(gamma)\tilde{\gamma}γ~ exists, the next and final step is not finding an algebraic approximation for a C∞C∞C^(oo)\mathscr{C}^{\infty}C∞ loop in a fibre of ppppp, but, rather, considering the algebraic curve BBBBB underlying ξ(S1)ξS1xi(S^(1))\xi\left(\mathbf{S}^{1}\right)ξ(S1), viewing γ~γ~tilde(gamma)\tilde{\gamma}γ~ as a C∞C∞C^(oo)\mathscr{C}^{\infty}C∞ section of the projection
and looking for an algebraic section of X×YB→BX×YB→BXxx_(Y)B rarr BX \times_{Y} B \rightarrow BX×YB→B approximating γ~γ~tilde(gamma)\tilde{\gamma}γ~. Thus, even when we start with just two real varieties XXXXX and YYYYY, we need to consider one-parameter algebraic families of fibres of ppppp, rather than single fibres.
(2) Consider the example where ppppp is the blow-up of a surface YYYYY at a real point bbbbb and γγgamma\gammaγ meets p−1(b)(R)p−1(b)(R)p^(-1)(b)(R)p^{-1}(b)(\mathbf{R})p−1(b)(R), transversally. Then for any γ~γ~tilde(gamma)\tilde{\gamma}γ~ sufficiently close to γγgamma\gammaγ in the Euclidean topology, the loop p∘γ~p∘γ~p@ tilde(gamma)p \circ \tilde{\gamma}p∘γ~ has to go through bbbbb. Hence ξξxi\xiξ has to be required to go through bbbbb for a loop γ~γ~tilde(gamma)\tilde{\gamma}γ~ as above to exist. Thus, a condition of weak approximation type must be considered in conjunction with Questions 3.1 (as was already noted by Bochnak and Kucharz [8]).
Let us now similarly contemplate a fibration argument in the context of Question 3.3 (2). We assume that X→fBX→fBXrarr"f"B\mathscr{X} \xrightarrow{f} BX→fB can be factored as X→pY→gBX→pY→gBXrarr"p"Yrarr"g"B\mathscr{X} \xrightarrow{p} \mathscr{Y} \xrightarrow{g} BX→pY→gB, where the variety YYY\mathscr{Y}Y is smooth and proper over RRR\mathbf{R}R, the morphism ppppp is dominant with rationally connected generic fibre, and ggggg is flat. Starting from a section s:P→Xs:P→Xs:P rarrXs: P \rightarrow \mathscr{X}s:P→X of fffff over PPPPP such that s|P(R)sP(R)s|_(P(R))\left.s\right|_{P(\mathbf{R})}s|P(R) can be extended to a C∞C∞C^(oo)\mathscr{C}^{\infty}C∞ section s′s′s^(')s^{\prime}s′ of f|X(R)fX(R)f|_(X(R))\left.f\right|_{\mathscr{X}(\mathbf{R})}f|X(R), a positive answer to Question 3.3 (2) for ggggg produces for us a section τÏ„tau\tauÏ„ of ggggg that extends p∘sp∘sp@sp \circ sp∘s. Let Z=p−1(τ(B))Z=p−1(Ï„(B))Z=p^(-1)(tau(B))\mathscr{Z}=p^{-1}(\tau(B))Z=p−1(Ï„(B)) and let h:Z→Bh:Z→Bh:Zrarr Bh: \mathscr{Z} \rightarrow Bh:Z→B denote the restriction of fffff. At this point, one would like to apply a positive answer to Question 3.3 (2) for hhhhh to obtain a section of hhhhh extending sssss, thus completing the argument, as Z⊆XZ⊆XZsubeX\mathscr{Z} \subseteq \mathscr{X}Z⊆X. In order to do so, one needs to know that s|P(R):P(R)→Z(R)sP(R):P(R)→Z(R)s|_(P(R)):P(R)rarrZ(R)\left.s\right|_{P(\mathbf{R})}: P(\mathbf{R}) \rightarrow \mathscr{Z}(\mathbf{R})s|P(R):P(R)→Z(R) can be extended to a C∞C∞C^(oo)\mathscr{C}^{\infty}C∞ section of h|Z(R):Z(R)→B(R)hZ(R):Z(R)→B(R)h|_(Z(R)):Z(R)rarr B(R)\left.h\right|_{\mathscr{Z}(\mathbf{R})}: \mathscr{Z}(\mathbf{R}) \rightarrow B(\mathbf{R})h|Z(R):Z(R)→B(R). However, the map h|Z(R)hZ(R)h|_(Z(R))\left.h\right|_{\mathscr{Z}(\mathbf{R})}h|Z(R) in general even fails
to be surjective. To correct this problem, one should require, at the very least, that τ(B(R))Ï„(B(R))tau(B(R))\tau(B(\mathbf{R}))Ï„(B(R)) approximate, in the Euclidean topology, the image of p∘s′:B(R)→Y(R)p∘s′:B(R)→Y(R)p@s^('):B(R)rarrY(R)p \circ s^{\prime}: B(\mathbf{R}) \rightarrow \mathscr{Y}(\mathbf{R})p∘s′:B(R)→Y(R). Thus, all in all, an approximation condition in the Euclidean topology has to be considered in conjunction with Question 3.3 (2).
The above discussion leads to the following definition. (This definition slightly differs from that given in [6], which considers the more general question of approximating holomorphic maps by algebraic ones, Ã la Runge, and which, as a consequence, is useful also for studying complex curves on complex varieties, without reference to the reals; however, all of the statements we make below are true with respect to either of the definitions.)
Definition 3.4. Let BBBBB be a smooth, proper, connected curve over R. A variety XXXXX over k=R(B)k=R(B)k=R(B)k=\mathbf{R}(B)k=R(B) satisfies the tight approximation property if for any proper model f:X→Bf:X→Bf:Xrarr Bf: \mathscr{X} \rightarrow Bf:X→B of XXXXX over BBBBB with XXX\mathscr{X}X smooth over RRR\mathbf{R}R, any reduced 0 -dimensional subscheme P⊂BP⊂BP sub BP \subset BP⊂B, any section s′:P→Xs′:P→Xs^('):P rarrXs^{\prime}: P \rightarrow \mathscr{X}s′:P→X of fffff over PPPPP and any C∞C∞C^(oo)\mathscr{C}^{\infty}C∞ section s:B(R)→X(R)s:B(R)→X(R)s:B(R)rarrX(R)s: B(\mathbf{R}) \rightarrow \mathscr{X}(\mathbf{R})s:B(R)→X(R) of f|X(R)fX(R)f|_(X(R))\left.f\right|_{\mathscr{X}(\mathbf{R})}f|X(R) such that s|P(R)=s′|P(R)sP(R)=s′P(R)s|_(P(R))=s^(')|_(P(R))\left.s\right|_{P(\mathbf{R})}=\left.s^{\prime}\right|_{P(\mathbf{R})}s|P(R)=s′|P(R), there exists a section σ:B→Xσ:B→Xsigma:B rarrX\sigma: B \rightarrow \mathscr{X}σ:B→X of fffff such that σ|P=s′|PσP=s′Psigma|_(P)=s^(')|_(P)\left.\sigma\right|_{P}=\left.s^{\prime}\right|_{P}σ|P=s′|P and such that σ|B(R)σB(R)sigma|_(B(R))\left.\sigma\right|_{B(\mathbf{R})}σ|B(R) lies arbitrarily close to sssss in the compact-open topology.
Given a smooth, proper, rationally connected variety XXXXX over RRR\mathbf{R}R, the validity of the tight approximation property for the variety obtained from XXXXX by extension of scalars from RRR\mathbf{R}R to R(t)R(t)R(t)\mathbf{R}(t)R(t) implies positive answers to Questions 3.1 for XXXXX.
The tight approximation property is (tautologically) a birational invariant, and it holds for PknPknP_(k)^(n)\mathbf{P}_{k}^{n}Pkn by a theorem of Bochnak and Kucharz [8]. (In [8], weak approximation conditions at complex points are ignored, but they create no additional difficulty.) The next two results provide more examples of varieties satisfying tight approximation.
Theorem 3.5. Let kkkkk be the function field of a real curve. Let XXXXX be a smooth variety over kkkkk. Let GGGGG be a linear algebraic group over kkkkk. Let f:Y→Xf:Y→Xf:Y rarr Xf: Y \rightarrow Xf:Y→X be a left torsor under GGGGG. Consider twists f′:Y′→Xf′:Y′→Xf^('):Y^(')rarr Xf^{\prime}: Y^{\prime} \rightarrow Xf′:Y′→X of fffff by right torsors under GGGGG, over kkkkk. If every such Y′Y′Y^(')Y^{\prime}Y′ satisfies the tight approximation property, then so does XXXXX.
Theorem 3.6. Let kkkkk be the function field of a real curve. Let p:Z→Bp:Z→Bp:Z rarr Bp: Z \rightarrow Bp:Z→B be a dominant morphism between smooth varieties over kkkkk. If BBBBB and the fibres of ppppp above the rational points of a dense open subset of B satisfy the tight approximation property, then so does ZZZZZ.
3.5. Homogeneous spaces
We are now in a position to sketch the proof of the following theorem, which in the "constant case," i.e., when the algebraic group and the homogeneous space are both defined over RRR\mathbf{R}R, immediately implies Theorem B.
Theorem 3.7. Homogeneous spaces of connected linear algebraic groups over the function field of a real curve satisfy the tight approximation property.
The proof of Theorem 3.7 starts by noting that quasitrivial tori over kkkkk are kkkkk-rational, hence satisfy the tight approximation property (since so does PknPknP_(k)^(n)\mathbf{P}_{k}^{n}Pkn ). Any torus TTTTT can be inserted into an exact sequence 1→S→Q→T→11→S→Q→T→11rarr S rarr Q rarr T rarr11 \rightarrow S \rightarrow Q \rightarrow T \rightarrow 11→S→Q→T→1 where SSSSS is a torus and QQQQQ is a quasitrivial torus. As any twist of QQQQQ as a torsor remains isomorphic to QQQQQ (Hilbert's Theorem 90) and hence satisfies the tight approximation property, we deduce, by the descent method (Theorem 3.5), that all tori over kkkkk satisfy the tight approximation property. Next, as every connected linear algebraic group over kkkkk is birationally equivalent to a relative torus over a kkkkk-rational variety (namely over the variety of maximal tori, when the algebraic group is reductive), we deduce, by the fibration method (Theorem 3.6), that connected linear algebraic groups over kkkkk satisfy the tight approximation property. By descent (Theorem 3.5 again), it follows that homogeneous spaces of connected linear algebraic groups over kkkkk satisfy the tight approximation property when they have a rational point. Finally, it is a theorem of Scheiderer that homogeneous spaces of connected linear algebraic groups over kkkkk satisfy the Hasse principle with respect to the real closures of kkkkk, so that if XXXXX denotes such a homogeneous space, then X(k)≠∅X(k)≠∅X(k)!=O/X(k) \neq \varnothingX(k)≠∅ whenever a C∞C∞C^(oo)\mathscr{C}^{\infty}C∞ section s:B(R)→X(R)s:B(R)→X(R)s:B(R)rarrX(R)s: B(\mathbf{R}) \rightarrow \mathscr{X}(\mathbf{R})s:B(R)→X(R) as in Definition 3.4 exists. This completes the proof of Theorem 3.7.
The main open problem surrounding the notion of tight approximation is the following.
Question 3.8. Let kkkkk be the function field of a real curve. Do all rationally connected varieties over kkkkk satisfy the tight approximation property?
Building on Theorems 3.5 and 3.6, the tight approximation property is shown in [6] to hold for various classes of rationally connected varieties beyond homogeneous spaces of connected linear algebraic groups. For instance, it holds for smooth cubic hypersurfaces
of dimension ≥2≥2>= 2\geq 2≥2 that are defined over RRR\mathbf{R}R, thus yielding, for such hypersurfaces, a positive answer to (the second part of) Question 3.1 (1).
Question 3.8 is open for cubic surfaces over kkkkk. Even Question 3.3 (2) is open when XXXXX is a cubic surface, although Question 3.3 (1) has an affirmative answer in this case, by a theorem of Tian [99].
Naturally, one hopes for the answer to Question 3.8 to be in the affirmative in general. This conjecture would have a host of interesting consequences, among which: a version of the Graber-Harris-Starr theorem over the reals (i.e., a positive answer to Question 3.3 (2) when P=∅P=∅P=O/P=\varnothingP=∅ ); Lang's widely open conjecture from [70] that the function field of a real curve with no real point is C1C1C_(1)C_{1}C1 (see [55, coRoLLARY 1.5] for the implication); and the existence of a geometrically rational curve on any smooth, proper, rationally connected variety of dimension ≥1≥1>= 1\geq 1≥1 over RRR\mathbf{R}R.
This last consequence is a conjecture of Kollár, who showed the existence of rational curves on those real rationally connected varieties of dimension ≥1≥1>= 1\geq 1≥1 that have real points (see [2, Remarks 20]). For real rationally connected varieties with no real point, it is interesting to consider a weaker property: the existence of a geometrically irreducible curve of even geometric genus. The latter can be reinterpreted in terms of the real integral Hodge conjecture (see [4]). Using Hodge theory and a real adaptation of Green's infinitesimal criterion for the density of Noether-Lefschetz loci, such curves of even genus can be shown to exist on all real Fano threefolds (see [5]). However, even on smooth quartic hypersurfaces in PR4PR4P_(R)^(4)\mathbf{P}_{\mathbf{R}}^{4}PR4, the existence of geometrically rational curves remains a challenge, as well as the mere existence of an absolute bound, independent of the chosen quartic hypersurface, on the minimal geometric genus of a geometrically irreducible curve of even geometric genus lying on such a hypersurface.
4. FUNCTION FIELDS OF CURVES OVER ppppp-ADIC FIELDS
4.1. Some motivation: rational curves over number fields
Even though the main questions about rational points of rationally connected varieties over number fields and over function fields of real curves are still wide open, the
Brauer-Manin obstruction and the tight approximation property at least provide rather satisfactory conjectural answers. It would be highly desirable to obtain a similar conjectural picture for rational points over other fields, for significant classes of varieties-including, at a minimum, concrete criteria for the existence of rational points.
Over the field Q(t)Q(t)Q(t)\mathbf{Q}(t)Q(t), this would encompass questions about rational curves on rationally connected varieties over QQQ\mathbf{Q}Q, about which very little is known. For example, it is unknown whether any rationally connected variety of dimension ≥1≥1>= 1\geq 1≥1 over QQQ\mathbf{Q}Q that possesses a rational point also contains a rational curve defined over QQQ\mathbf{Q}Q. Much more ambitiously, it is unknown whether any such variety contains enough rational curves to imply the finiteness of the set of RRRRR-equivalence classes of rational points, a question asked in [16, QuEsTION 10.12]. (Known results on this problem are listed after Question 10.12 in [16].) As another example, the regular inverse Galois problem over QQQ\mathbf{Q}Q, which asks for the construction of a regular Galois extension of Q(t)Q(t)Q(t)\mathbf{Q}(t)Q(t) with specified Galois group, and which can be reinterpreted as a problem about the existence of appropriate rational curves on the homogeneous space SLn/GSLn/GSL_(n)//G\mathrm{SL}_{n} / GSLn/G over QQQ\mathbf{Q}Q is open even for finite nilpotent groups GGGGG. All of these problems are currently out of reach.
As a first step towards these questions, let us replace QQQ\mathbf{Q}Q with its completions and turn to rational points over the field Qp(t)Qp(t)Q_(p)(t)\mathbf{Q}_{p}(t)Qp(t) or over its finite extensions.
4.2. Rational curves on varieties over ppppp-adic fields
In the constant case (that is, for varieties obtained by scalar extension from varieties defined over a ppppp-adic field, i.e., a finite extension of QpQpQ_(p)\mathbf{Q}_{p}Qp ), various existence results are known:
(2) for any smooth, proper, rationally connected variety XXXXX over a ppppp-adic field kkkkk, Kollár [64,66][64,66][64,66][64,66][64,66] has shown that the rational points of XXXXX fall into finitely many RRRRR-equivalence classes, and that there exist rational curves on XXXXX, defined over kkkkk, passing through any finite set of rational points of XXXXX that belong to the same RRRRR-equivalence class (with prescribed jets of any given order at these points).
This last statement concerns conditions of weak approximation type that can be imposed on rational curves on rationally connected varieties over ppppp-adic fields. It would be interesting to formulate an analogue, in this ppppp-adic context, of the surjectivity of the Borel-Haefliger cycle class map CH1(X)→H1(X(R),Z/2Z)CH1(X)→H1(X(R),Z/2Z)CH_(1)(X)rarrH_(1)(X(R),Z//2Z)\mathrm{CH}_{1}(X) \rightarrow H_{1}(X(\mathbf{R}), \mathbf{Z} / 2 \mathbf{Z})CH1(X)→H1(X(R),Z/2Z) (i.e., of Questions 3.1 (2)).
We saw in Section 3 that in order to answer questions about homology classes of rational curves on real varieties, it can be useful to consider more generally the tight approximation property, for nonconstant varieties over the function field of a real curve. By analogy, this gives incentive to investigate the possibility of a ppppp-adic analogue of the tight approximation property for nonconstant varieties over the function field of a curve over a ppppp-adic field, the validity of which would have consequences for a likely easier to formulate ppppp-adic integral Hodge conjecture for 1-cycles on varieties over ppppp-adic fields.
4.3. Quadrics and other homogeneous spaces
In the nonconstant case, even the simplest varieties over Qp(t)Qp(t)Q_(p)(t)\mathbf{Q}_{p}(t)Qp(t) lead to difficult problems when it comes to their rational points. For instance, it is only a relatively recent theorem of Parimala and Suresh [86], for p≠2p≠2p!=2p \neq 2p≠2, and of Leep [72], based on work of Heath-Brown [54], for arbitrary ppppp, that every projective quadric of dimension ≥7≥7>= 7\geq 7≥7 over Qp(t)Qp(t)Q_(p)(t)\mathbf{Q}_{p}(t)Qp(t) possesses a rational point. (In the language of quadratic forms, "the uuuuu-invariant of Qp(t)Qp(t)Q_(p)(t)\mathbf{Q}_{p}(t)Qp(t) is equal to 8.") Many other articles have been devoted to local-global principles for varieties over function fields of curves over ppppp-adic fields (e.g., [19-21, 23, 24,37,38,40-46,56-58,77,85,87,88,92,98]).
A patching technique was developed by Harbater, Hartmann and Krashen ("patching over fields," a successor to formal patching), and was applied to study rational points of homogeneous spaces over such fields. It was used, in [42], to give another proof of the aforementioned theorem of Parimala and Suresh, and, in [23], to establish, more generally, the local-global principle for the existence of rational points on smooth projective quadrics of dimension ≥1≥1>= 1\geq 1≥1 over Qp(t)Qp(t)Q_(p)(t)\mathbf{Q}_{p}(t)Qp(t) (or over a finite extension of Qp(t)Qp(t)Q_(p)(t)\mathbf{Q}_{p}(t)Qp(t) ), with respect to all discrete valuations on this field, when ppppp is odd.
Our goal is thus to define, in complete generality, a closed subset X(Ak)rec⊆X(Ak)XAkrec⊆XAkX(A_(k))^(rec)sube X(A_(k))X\left(\mathbf{A}_{k}\right)^{\mathrm{rec}} \subseteq X\left(\mathbf{A}_{k}\right)X(Ak)rec⊆X(Ak) containing X(k)X(k)X(k)X(k)X(k), using on the one hand a reciprocity law coming from kkkkk and on the other hand an analogue of the Brauer group of XXXXX.
Grothendieck's purity theorem for the Brauer group equates Br(X)Brâ¡(X)Br(X)\operatorname{Br}(X)Brâ¡(X) with the unramified cohomology group Hnr2(X/k,Q/Z(1))Hnr2(X/k,Q/Z(1))H_(nr)^(2)(X//k,Q//Z(1))H_{\mathrm{nr}}^{2}(X / k, \mathbf{Q} / \mathbf{Z}(1))Hnr2(X/k,Q/Z(1)). We recall the definition of unramified cohomology: for any irreducible smooth variety VVVVV over a field KKKKK of characteristic 0 and any torsion Galois module MMMMM over KKKKK, the group Hnrq(V/K,M)Hnrq(V/K,M)H_(nr)^(q)(V//K,M)H_{\mathrm{nr}}^{q}(V / K, M)Hnrq(V/K,M) is the subgroup of the Galois cohomology group Hq(K(V),M)Hq(K(V),M)H^(q)(K(V),M)H^{q}(K(V), M)Hq(K(V),M) consisting of those classes whose residues along all codimension 1 points of VVVVV vanish. It is the unramified cohomology group Hnr3(X/k,Q/Z(2))Hnr3(X/k,Q/Z(2))H_(nr)^(3)(X//k,Q//Z(2))H_{\mathrm{nr}}^{3}(X / k, \mathbf{Q} / \mathbf{Z}(2))Hnr3(X/k,Q/Z(2)) that will serve as a substitute for Br(X)Brâ¡(X)Br(X)\operatorname{Br}(X)Brâ¡(X) here. (The shift in degree is explained by the fact that the field kkkkk has cohomological dimension 3 while number fields have virtual cohomological dimension 2.) For any field extension K/kK/kK//kK / kK/k, Bloch-Ogus theory provides an evaluation map Hnr3(X/k,Q/Z(2))→H3(K,Q/Z(2)),α↦α(x)Hnr3(X/k,Q/Z(2))→H3(K,Q/Z(2)),α↦α(x)H_(nr)^(3)(X//k,Q//Z(2))rarrH^(3)(K,Q//Z(2)),alpha|->alpha(x)H_{\mathrm{nr}}^{3}(X / k, \mathbf{Q} / \mathbf{Z}(2)) \rightarrow H^{3}(K, \mathbf{Q} / \mathbf{Z}(2)), \alpha \mapsto \alpha(x)Hnr3(X/k,Q/Z(2))→H3(K,Q/Z(2)),α↦α(x) along any KKKKK-point xxxxx of XXXXX (see [7]).
Let BBB\mathscr{B}B denote an irreducible normal proper scheme over ZpZpZ_(p)\mathbf{Z}_{p}Zp with function field kkkkk. In contrast with what happens over number fields, here it is not one reciprocity law that will play a rôle, but infinitely many of them: one for each closed point of BBB\mathscr{B}B, for each such BBB\mathscr{B}B. Namely, given any closed point b∈Bb∈Bb inBb \in \mathscr{B}b∈B, Kato [61,§1][61,§1][61,§1][61, \S 1]§[61,§1] has constructed a complex
where ξξxi\xiξ ranges over the set B1,bB1,bB_(1,b)\mathscr{B}_{1, b}B1,b of 1-dimensional irreducible closed subsets of BBB\mathscr{B}B that contain bbbbb, and where κ(ξ)κ(ξ)kappa(xi)\kappa(\xi)κ(ξ) denotes the function field of ξξxi\xiξ (which is either a global field of characteristic ppppp or a local field of characteristic 0 ). The second arrow in (4.1) is the sum of the invariant maps from local class field theory at the finitely many places of κ(ξ)κ(ξ)kappa(xi)\kappa(\xi)κ(ξ) that lie over bbbbb. The first arrow of (4.1) is induced by residue maps ∂v:H3(kv,Q/Z(2))→Br(κ(ξ))∂v:H3kv,Q/Z(2)→Brâ¡(κ(ξ))del_(v):H^(3)(k_(v),Q//Z(2))rarr Br(kappa(xi))\partial_{v}: H^{3}\left(k_{v}, \mathbf{Q} / \mathbf{Z}(2)\right) \rightarrow \operatorname{Br}(\kappa(\xi))∂v:H3(kv,Q/Z(2))→Brâ¡(κ(ξ)) constructed by Kato in [61], where vvvvv denotes the discrete valuation of kkkkk defined by ξξxi\xiξ.
Although evidence is scarce, the answer to the following question might always be in the affirmative, as far as one knows:
Question 4.1. Let kkkkk be a finite extension of Qp(t)Qp(t)Q_(p)(t)\mathbf{Q}_{p}(t)Qp(t). Let XXXXX be a smooth, proper, rationally connected variety over kkkkk. If X(Ak)rec ≠∅XAkrec ≠∅X(A_(k))^("rec ")!=O/X\left(\mathbf{A}_{k}\right)^{\text {rec }} \neq \varnothingX(Ak)rec ≠∅, does it follow that X(k)≠∅X(k)≠∅X(k)!=O/X(k) \neq \varnothingX(k)≠∅ ?
Question 4.1 has a positive answer when XXXXX is a quadric and p≠2p≠2p!=2p \neq 2p≠2. Indeed, we recall from Section 4.3 that even X(Ak)≠∅XAk≠∅X(A_(k))!=O/X\left(\mathbf{A}_{k}\right) \neq \varnothingX(Ak)≠∅ then implies X(k)≠∅X(k)≠∅X(k)!=O/X(k) \neq \varnothingX(k)≠∅ (see [23]). It also has a positive answer when XXXXX is birationally equivalent to a torsor under a torus over kkkkk. This follows from the work of Harari, Scheiderer, Szamuely, Tian [38, THEOREM 5.1], [97, § 0.3.1] (modulo the comparison between the reciprocity obstruction defined here and the reciprocity obstruction considered in these articles; the latter is weaker, but turns out to suffice to detect rational points on torsors under tori). We note that there are examples of torsors under tori over kkkkk whose smooth compactifications XXXXX satisfy X(Ak)rec =∅XAkrec =∅X(A_(k))^("rec ")=O/X\left(\mathbf{A}_{k}\right)^{\text {rec }}=\varnothingX(Ak)rec =∅ while X(Ak)≠∅XAk≠∅X(A_(k))!=O/X\left(\mathbf{A}_{k}\right) \neq \varnothingX(Ak)≠∅ (see [24, RemarQUE 5.10]). Positive answers to Question 4.1 are known in various other cases in which XXXXX is birationally equivalent to a homogeneous space of a connected linear algebraic group over kkkkk. For specific statements, we refer the reader to the articles quoted in Section 4.3. Question 4.1 remains open in general for smooth compactifications of torsors under connected linear alge-
braic groups over kkkkk, for smooth compactifications of homogeneous spaces of SLnSLnSL_(n)\mathrm{SL}_{n}SLn with finite stabilisers, and for conic bundle surfaces over Pk1Pk1P_(k)^(1)\mathbf{P}_{k}^{1}Pk1.
Question 4.1 focuses on the existence of rational points rather than on the density of X(k)X(k)X(k)X(k)X(k) in X(Ak)rec XAkrec X(A_(k))^("rec ")X\left(\mathbf{A}_{k}\right)^{\text {rec }}X(Ak)rec as the latter property is only known for projective space (see [3, тHEoREM 1]) and hence for varieties that are rational as soon as they possess a rational point, such as quadrics. For smooth compactifications of tori, the density of X(k)X(k)X(k)X(k)X(k) in X(Ak)rec XAkrec X(A_(k))^("rec ")X\left(\mathbf{A}_{k}\right)^{\text {rec }}X(Ak)rec is known to hold off the set of discrete valuations of kkkkk whose residue field has characteristic ppppp (see [37, THEOREM 5.2]; for the meaning of "off" here, see [103, DEFINITION 2.9]).
To obtain more positive answers to Question 4.1, it is natural to wish for flexible tools such as general descent theorems and fibration theorems. In the same way that introducing the tight approximation property and replacing Question 3.3 (2) with Question 3.8 was a key step to obtain a problem that behaves well with respect to fibrations into rationally connected varieties (see the discussion in Section 3.2), it is likely that in order to obtain compatibility with descent and fibrations, one will have to strengthen Question 4.1 by incorporating into it a ppppp-adic analogue of the approximation condition in the Euclidean topology that appears in Definition 3.4. The main challenge, here, is to provide the correct formulation for such a ppppp-adic tight approximation property.
We note that in any case, a general fibration theorem has to lie deep, as it would presumably give a direct route to the local-global principle for the existence of rational points on smooth projective quadrics over kkkkk (so far unknown when p=2p=2p=2p=2p=2 ) and hence to the computation of the uuuuu-invariant of kkkkk (equal to 8 ; see Section 4.3). Indeed, in the case of conics over kkkkk, this local-global principle follows from Tate-Lichtenbaum duality [73]; applying a fibration theorem to a general pencil of hyperplane sections of a fixed smooth projective quadric of dimension n≥2n≥2n >= 2n \geq 2n≥2 would allow one to deduce the general case by induction on nnnnn.
4.6. Further questions
A good understanding of rational points of rationally connected varieties over function fields of curves over ppppp-adic fields, be it via Question 4.1 or otherwise, should shed light on concrete test questions such as the following:
Questions 4.2. Let ppppp be a prime number and kkkkk be a finite extension of Qp(t)Qp(t)Q_(p)(t)\mathbf{Q}_{p}(t)Qp(t).
(1) Does the conjecture of MinÃ¡Ä and Tân on the vanishing of Massey products in Galois cohomology hold for kkkkk ? (See Section 2.5 and [78,79][78,79][78,79][78,79][78,79].)
(2) Is there an algorithm that takes as input a smooth, projective, rationally connected variety XXXXX over kkkkk and decides whether XXXXX has a rational point?
One might approach the first of these questions by trying to mimic [49] over kkkkk, which would require making progress on the arithmetic, over kkkkk, of homogeneous spaces of SLnSLnSL_(n)\mathrm{SL}_{n}SLn with finite supersolvable geometric stabilisers.
To put the second question in perspective, let us recall what is known about algorithms for deciding the existence of rational points on arbitrary varieties ("Hilbert's tenth problem") over various fields of interest. Over QQQ\mathbf{Q}Q or C(t)C(t)C(t)\mathbf{C}(t)C(t), the existence of such an algorithm
is an outstanding open problem. Denef [29] showed that over R(t)R(t)R(t)\mathbf{R}(t)R(t), such an algorithm does not exist. His method was extended to prove that there is no such algorithm over Qp(t)Qp(t)Q_(p)(t)\mathbf{Q}_{p}(t)Qp(t) (Kim and Roush [62], completed by Degroote and Demeyer [27]), over any finite extension of R(t)R(t)R(t)\mathbf{R}(t)R(t) that possesses a real place (Moret-Bailly [81]), or, when p≠2p≠2p!=2p \neq 2p≠2, over any finite extension of Qp(t)Qp(t)Q_(p)(t)\mathbf{Q}_{p}(t)Qp(t) (Eisenträger [32], Moret-Bailly [81]). In addition, over number fields, it is known that restricting from arbitrary varieties to smooth projective varieties makes no difference (see [96, § II.7], [90, THEOREM 1.1 (I)]). Restricting to smooth, projective, rationally connected varieties, however, does make a drastic difference: Question 4.2 (2) might well have an affirmative answer for all of the fields just mentioned. Over C(t)C(t)C(t)\mathbf{C}(t)C(t), this is trivially so, by the Graber-Harris-Starr theorem. Over R(t)R(t)R(t)\mathbf{R}(t)R(t), a positive answer to Question 4.2 (2) would follow from a positive answer to Question 3.8. Indeed, in the notation of Definition 3.4, if XXXXX satisfies the tight approximation property, then XXXXX has a rational point if and only if f|X(R)fX(R)f|_(X(R))\left.f\right|_{\mathscr{X}(\mathbf{R})}f|X(R) admits a C∞C∞C^(oo)\mathscr{C}^{\infty}C∞ section, a property that can be decided algorithmically. Over number fields, as was observed by Poonen [89, Remark 5.3], a positive answer to Question 4.2 (2) would follow from the conjecture that rational points are always dense in the Brauer-Manin set. It seems likely that a positive answer to Question 4.1 would similarly imply a positive answer to Question 4.2 (2). To mimic Poonen's argument, one runs into the difficulty that the elements of Hnr3(X/k,Q/Z(2))Hnr3(X/k,Q/Z(2))H_(nr)^(3)(X//k,Q//Z(2))H_{\mathrm{nr}}^{3}(X / k, \mathbf{Q} / \mathbf{Z}(2))Hnr3(X/k,Q/Z(2)) are harder to describe than those of Hnr2(X/k,Q/Z(1))=Br(X)Hnr2(X/k,Q/Z(1))=Brâ¡(X)H_(nr)^(2)(X//k,Q//Z(1))=Br(X)H_{\mathrm{nr}}^{2}(X / k, \mathbf{Q} / \mathbf{Z}(1))=\operatorname{Br}(X)Hnr2(X/k,Q/Z(1))=Brâ¡(X), whose interpretation in terms of Azumaya algebras is a key point in [89, REMARK 5.3]; however, this can be remedied by viewing Hnr3(X/k,Q/Z(2))Hnr3(X/k,Q/Z(2))H_(nr)^(3)(X//k,Q//Z(2))H_{\mathrm{nr}}^{3}(X / k, \mathbf{Q} / \mathbf{Z}(2))Hnr3(X/k,Q/Z(2)), using Bloch-Ogus theory, as the group of
global sections of the Zariski sheaf associated with the presheaf U↦Het3(U,Q/Z(2))U↦Het3(U,Q/Z(2))U|->H_(et)^(3)(U,Q//Z(2))U \mapsto H_{\mathrm{et}}^{3}(U, \mathbf{Q} / \mathbf{Z}(2))U↦Het3(U,Q/Z(2)), and describing Het3(U,Q/Z(2))Het3(U,Q/Z(2))H_(et)^(3)(U,Q//Z(2))H_{\mathrm{et}}^{3}(U, \mathbf{Q} / \mathbf{Z}(2))Het3(U,Q/Z(2)) via ÄŒech cohomology.
4.7. Other fields
There are a number of other fields over which a better understanding of rational points of rationally connected varieties would be valuable. One of the simplest example is the fraction field k=C((x,y))k=C((x,y))k=C((x,y))k=\mathbf{C}((x, y))k=C((x,y)) of the ring of formal power series C[[x,y]]C[[x,y]]C[[x,y]]\mathbf{C}[[x, y]]C[[x,y]], which can be seen as a first step before considering function fields of complex surfaces. This field presents both local and global features, and a reciprocity obstruction can again be defined (in terms of the unramified Brauer group-recall that kkkkk has cohomological dimension 2). This obstruction was used in [24] to produce the first example of a torsor YYYYY under a torus, over kkkkk, such that Y(k)=∅Y(k)=∅Y(k)=O/Y(k)=\varnothingY(k)=∅ but Y(kv)≠∅Ykv≠∅Y(k_(v))!=O/Y\left(k_{v}\right) \neq \varnothingY(kv)≠∅ for every discrete valuation vvvvv on kkkkk. The analogues of Question 4.1 and of Questions 4.2 can be asked over this field, too. It is not known, however, whether the reciprocity obstruction explains the absence of rational points on smooth proper varieties that are birationally equivalent to torsors under tori over kkkkk (though see [59, CORoLLAIRE 4.4] for a closely related result involving possibly ramified Brauer classes). We refer the interested reader to [18,22,59,60][18,22,59,60][18,22,59,60][18,22,59,60][18,22,59,60] for the state of the art.
Benoist and Yonatan Harpaz for the pleasant collaborations that have led to the results reported on in this article.
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3 In [32], Paris showed that ItPH23ItPH23ItPH_(2)^(3)\mathrm{ItPH}_{2}^{3}ItPH23 is independent of PAPAPA\mathrm{PA}PA, while his argument implies the equivalence of statements 2 and 3. See Section 5.2.
5 See Section 5 for the construction of the moduli space MdNMdNM_(d)^(N)\mathcal{M}_{d}^{N}MdN.
6 Although in fairness it should be noted that [91] suggests the opposite conclusion, stating: "Are there any rational periodic orbits of a quadratic x2+cx2+cx^(2)+cx^{2}+cx2+c of period greater than 3? The results for periods 1,2 , and 3 would lead one to suspect that there must be."
7 We remark that it is easy to prove uniform boundedness for xd+cxd+cx^(d)+cx^{d}+cxd+c over QQQ\mathbb{Q}Q when ddddd is odd, and more generally over any field K/QK/QK//QK / \mathbb{Q}K/Q with a real embedding. Indeed, it is an elementary fact that if f:R→Rf:R→Rf:RrarrRf: \mathbb{R} \rightarrow \mathbb{R}f:R→R is any nondecreasing function, then fffff has no nonfixed periodic points; cf. [64].
8 The gonality of an algebraic curve XXXXX, or its function field, is the minimal degree of a nonconstant map X→P1X→P1X rarrP^(1)X \rightarrow \mathbb{P}^{1}X→P1.
1 Of course, the following discussion also applies to U(W)U(W)U(W)U(W)U(W).
2 A Whittaker datum of U(V′)UV′U(V^('))U\left(V^{\prime}\right)U(V′) is a pair (N,θ)(N,θ)(N,theta)(N, \theta)(N,θ) consisting of a maximal unipotent subgroup
N⊂U(V′)N⊂UV′N sub U(V^('))N \subset U\left(V^{\prime}\right)N⊂U(V′) and a generic character θ:N→C×θ:N→C×theta:N rarrC^(xx)\theta: N \rightarrow \mathbb{C}^{\times}θ:N→C×. This datum only matters up to conjugacy.
1 We make a small abuse of notation by using XXXXX to denote both the conjugacy class from the introduction, which is used in the definition of a Shimura datum, and the quotient G(R)/K∞∘A∞∘G(R)/K∞∘A∞∘G(R)//K_(oo)^(@)A_(oo)^(@)G(\mathbb{R}) / K_{\infty}^{\circ} A_{\infty}^{\circ}G(R)/K∞∘A∞∘ considered in this section. See [24$§2.4][24$§2.4][24$§2.4][\mathbf{2 4} \mathbf{\$} \mathbf{\S} \mathbf{2 . 4}]§[24$§2.4] for an extended discussion of the various quotients.
3 As a consequence of the comparison with moduli spaces of local shtukas in [56], one obtains a group-theoretic characterization of Rapoport-Zink spaces as local Shimura varieties determined by the tuple (G,b,μ)(G,b,μ)(G,b,mu)(G, b, \mu)(G,b,μ). We suppress (G,μ)(G,μ)(G,mu)(G, \mu)(G,μ) from the notation for simplicity.
4 The result precedes the notion of diamonds and, in order to ensure that SKp∘bSKp∘bS_(K^(p))^(@b)S_{K^{p}}^{\circ b}SKp∘b is a diamond, one needs to take care in defining it. At hyperspecial level, one should consider the adic generic fiber of the formal completion of the integral model of the Shimura variety along the Newton stratum indexed by bbbbb in its special fiber.
2 See [125] and [81] for definition and properties of expanders.
3 Let ppppp be a large prime and denote by X∗(p)=X(p)∖(0,0,0)X∗(p)=X(p)∖(0,0,0)X^(**)(p)=X(p)\\(0,0,0)X^{*}(p)=X(p) \backslash(0,0,0)X∗(p)=X(p)∖(0,0,0) the solutions of (1.1) modulo ppppp with the removal of (0,0,0)(0,0,0)(0,0,0)(0,0,0)(0,0,0). The Markoff graphs are obtained by joining each xxxxx in X∗(p)X∗(p)X^(**)(p)X^{*}(p)X∗(p) to Rj(x),j=1,2,3Rj(x),j=1,2,3R_(j)(x),j=1,2,3R_{j}(x), j=1,2,3Rj(x),j=1,2,3. They were considered first by Arthur Baragar in his thesis [3].
6 The exponent 1313(1)/(3)\frac{1}{3}13 in (1.5) has been improved to 7979(7)/(9)\frac{7}{9}79 in [87].
7 We remark that in [52] Corvaja and Zannier showed that the greatest prime factor of xyxyxyx yxy for a Markoff triple (x,y,z)(x,y,z)(x,y,z)(x, y, z)(x,y,z) tends to infinity.
8 "Trotz der außerordentlich merkwürdigen und wichtigen Resultate scheinen diese schwiergen Untersuchungen wenig bekannt zu sein" [In spite of the extraordinarily noteworthy and important results these difficult investigations seem to be little known]
3 This argument can be used to construct an Eichler-Shimura isomorphism in families for GSp4GSp4GSp_(4)\mathrm{GSp}_{4}GSp4, which interpolates the classical H1H1H^(1)H^{1}H1 comparison isomorphism at almost all classical points - see [51].
3 Sometimes these methods extend beyond the stated hypotheses. For example, [18] also works for Kisin-Pappas models (Section 6), and [26] proves a uniformization result also in the nonbasic case. However, we will not try to present the methods in their maximally general settings.
2 If char F=F=F=F=F= char kkkkk (the equal characteristic case), this assumption on kkkkk is not necessary. We impose it here to have a uniform treatment of both equal and mixed characteristic (i.e., char F≠chark)F≠charâ¡k)F!=char k)F \neq \operatorname{char} k)F≠charâ¡k) cases. For the same reason, we work with perfect algebraic geometry below even in equal characteristic.
5 Such restriction functor defines the so-called fusion product, a key concept in the geometric Satake equivalence. The terminology "fusion" originally comes from conformal field theory.
1 Voevodsky showed this is not possible integrally, so the best one can hope for is a ttttt structure with QQQ\mathbb{Q}Q-coefficients.
2 In fact, at the beginning of §3§3§3\S 3§§3 of [28], Bloch and Kato write, "The cohomological symbol defined by Tate [114] gives a map . . . which one conjectures to be an isomorphism quite generally."
1 For example, the square-tiled surface in Figure 1 is made up of 54 squares, has 3 conical points of angle 3π3Ï€3pi3 \pi3Ï€ (corresponding to simple zeros of qqqqq ), and 7 conical points of angle πÏ€pi\piÏ€ (corresponding to simple poles of qqqqq ). Therefore, it has genus 0 and belongs to the principal stratum Q(13,−17)Q13,−17Q(1^(3),-1^(7))\mathcal{Q}\left(1^{3},-1^{7}\right)Q(13,−17).
1 In the classical formulation, Tian's CM weight or, equivalently, the Donaldson-Futaki invariant is used to define the K-stability. However, to fit our discussion in the nonArchimedean framework, we use the equivalent formulation via the MNAMNAM^(NA)\mathbf{M}^{\mathrm{NA}}MNA functional.